From dd3a13fc795c962e0e267c5a552f74d85853da93 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Tue, 9 Mar 2021 14:35:55 +0100 Subject: [PATCH 01/30] feat: fixed notebook with resultant matrix --- "Mermin\342\200\223Peres-Game.ipynb" | 260 +++++++++++++++++++++++++++ README.md | 10 +- 2 files changed, 269 insertions(+), 1 deletion(-) create mode 100644 "Mermin\342\200\223Peres-Game.ipynb" diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" new file mode 100644 index 0000000..f6dc571 --- /dev/null +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -0,0 +1,260 @@ +{ + "metadata": { + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.1-final" + }, + "orig_nbformat": 2, + "kernelspec": { + "name": "python3", + "display_name": "Python 3", + "language": "python" + } + }, + "nbformat": 4, + "nbformat_minor": 2, + "cells": [ + { + "source": [ + "# Mermin Peres Magic Square Game" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "from qiskit import *\n", + "import numpy as np" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "def check_result(results, col, row):\n", + "\n", + " for i in results: \n", + " i = np.array(list(i.split())).astype(np.int)[::-1]\n", + " # bring array in correct i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", + " global a\n", + " a = i[0:2]\n", + " a_sum = np.sum(a)\n", + " a3 = int(np.mod(a_sum+1,2))\n", + " a = np.append(a,a3)\n", + " winning_alice = np.mod(a_sum + a3, 2) == 1\n", + " # compute x3 by x1+x2+1 mod 2 and save to array\n", + "\n", + " b = i[2:]\n", + " b_sum = np.sum(b)\n", + " b3 = int(np.mod(b_sum,2))\n", + " b = np.append(b,b3)\n", + " winning_bob = np.mod(b_sum + b3, 2) == 0 \n", + " # compute y3 by y1+y2 mod 2 and save to array\n", + "\n", + " # compute overall winning condition -> alice at index of bob's row and vice versa\n", + " if(winning_bob & winning_alice & (a[row-1]==b[col-1])):\n", + " print(\"Victory!\") \n", + " print(\"The following magic square wins the game!\")\n", + " k = np.empty((3,3,))\n", + " k[:] = np.nan\n", + " k[:,col-1] = a\n", + " k[row-1,:] = b\n", + " print(k)\n", + "\n", + " else:\n", + " \"The Game is lost!\"\n", + "\n", + " print(\"\\n\")\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "Alice and Bob, select a column or row value of 1, 2, or 3\n", + "Alice choice (column):\n", + "Bob choice (row):\n", + " ┌───┐ ░ \n", + "q7_0: ┤ H ├──■────────░─\n", + " ├───┤ │ ░ \n", + "q7_1: ┤ H ├──┼────■───░─\n", + " └───┘┌─┴─┐ │ ░ \n", + "q8_0: ─────┤ X ├──┼───░─\n", + " └───┘┌─┴─┐ ░ \n", + "q8_1: ──────────┤ X ├─░─\n", + " └───┘ ░ \n", + "c4: 1/══════════════════\n", + " \n", + "c5: 1/══════════════════\n", + " \n", + "c6: 1/══════════════════\n", + " \n", + "c7: 1/══════════════════\n", + " \n", + "Victory!\n", + "The following is the winning magic square!\n", + "[[ 1. 1. 0.]\n", + " [ 1. nan nan]\n", + " [ 1. nan nan]]\n", + "\n", + "\n", + "Victory!\n", + "The following is the winning magic square!\n", + "[[ 1. 1. 0.]\n", + " [ 0. nan nan]\n", + " [ 0. nan nan]]\n", + "\n", + "\n", + "Victory!\n", + "The following is the winning magic square!\n", + "[[ 1. 0. 1.]\n", + " [ 1. nan nan]\n", + " [ 1. nan nan]]\n", + "\n", + "\n", + "Victory!\n", + "The following is the winning magic square!\n", + "[[ 0. 1. 1.]\n", + " [ 1. nan nan]\n", + " [ 0. nan nan]]\n", + "\n", + "\n", + "Victory!\n", + "The following is the winning magic square!\n", + "[[ 1. 0. 1.]\n", + " [ 0. nan nan]\n", + " [ 0. nan nan]]\n", + "\n", + "\n", + "Victory!\n", + "The following is the winning magic square!\n", + "[[ 0. 0. 0.]\n", + " [ 1. nan nan]\n", + " [ 0. nan nan]]\n", + "\n", + "\n", + "Victory!\n", + "The following is the winning magic square!\n", + "[[ 0. 1. 1.]\n", + " [ 0. nan nan]\n", + " [ 1. nan nan]]\n", + "\n", + "\n", + "Victory!\n", + "The following is the winning magic square!\n", + "[[ 0. 0. 0.]\n", + " [ 0. nan nan]\n", + " [ 1. nan nan]]\n", + "\n", + "\n" + ] + } + ], + "source": [ + "#The following function sets up the initial entanglement between Alice and Bob\n", + "def share_bell_state(qc,a,b,c,d): \n", + " qc.h(a)\n", + " qc.h(b)\n", + " qc.cx(a,c)\n", + " qc.cx(b,d)\n", + " \n", + "# The following functions represent the U(gamma) and V(gamma) controlled Cliffords.\n", + "def U(qc,gamma,a,b):\n", + " if gamma==1:\n", + " qc.h(a)\n", + " qc.i(b)\n", + " elif gamma==2:\n", + " qc.swap(a,b)#\n", + " qc.h(a)\n", + " qc.i(b)\n", + "# qc.swap(a,b)\n", + " elif gamma==3:\n", + " qc.cx(a,b)#\n", + " qc.h(a)\n", + " qc.i(b)\n", + "# qc.cx(a,b)\n", + "def V(qc,gamma,a,b):\n", + " if gamma==1:\n", + " qc.h(a)\n", + " qc.h(b)\n", + " if gamma==2:\n", + " qc.swap(a,b)\n", + " elif gamma==3:\n", + " qc.z(a)#\n", + " qc.z(b)#\n", + " qc.cz(a,b)#\n", + " qc.h(a)\n", + " qc.h(b)\n", + "# qc.cz(a,b)\n", + "# qc.z(a)\n", + "# qc.z(b)\n", + "\n", + "\n", + "#We will ask the user to select the row and column on Alice and Bob's behalf.\n", + "print(\"Alice and Bob, select a column or row value of 1, 2, or 3\") \n", + "print(\"Alice choice (column):\")\n", + "alpha = int(input())\n", + "print(\"Bob choice (row):\")\n", + "beta = int(input())\n", + "\n", + "#Create the quantum register and the classical register to store our final bit values\n", + "aliceQR = QuantumRegister(2)\n", + "x1CR = ClassicalRegister(1)\n", + "x2CR = ClassicalRegister(1)\n", + "bobQR = QuantumRegister(2)\n", + "y1CR = ClassicalRegister(1)\n", + "y2CR = ClassicalRegister(1)\n", + "\n", + "#Create the circuit\n", + "magicsquare_circuit = QuantumCircuit(aliceQR,bobQR,x1CR,x2CR,y1CR,y2CR)\n", + "\n", + "#Generate the Bell state on the circuit\n", + "share_bell_state(magicsquare_circuit,0,1,2,3)\n", + "\n", + "magicsquare_circuit.barrier()\n", + "magicsquare_circuit.draw(output='mpl')\n", + "print(magicsquare_circuit)\n", + "\n", + "#Draw the rest of the circuit based on Alice and Bob's selection \n", + "if 4>alpha>0 and 4>beta>0:\n", + " U(magicsquare_circuit,alpha,0,1) \n", + " V(magicsquare_circuit,beta,2,3)\n", + " magicsquare_circuit.measure(0,0)\n", + " magicsquare_circuit.measure(0,0)\n", + " magicsquare_circuit.measure(1,1)\n", + " magicsquare_circuit.measure(2,2)\n", + " magicsquare_circuit.measure(3,3)\n", + " \n", + " backend = BasicAer.get_backend('qasm_simulator') # define the backend\n", + " job = execute(magicsquare_circuit, backend, shots=100) # run the job simulation\n", + " results = job.result().get_counts()\n", + " check_result(results, alpha, beta)\n", + "\n", + " \n", + " \n", + "else:\n", + " print(\"please enter values of 1, 2, or 3\")" + ] + } + ] +} \ No newline at end of file diff --git a/README.md b/README.md index c1c9337..27a7a58 100644 --- a/README.md +++ b/README.md @@ -7,6 +7,7 @@ 4. GHZ Game [![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/JanLahmann/Fun-with-Quantum/master?filepath=GHZ-Game.ipynb) 5. GHZ Game on real devices [![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/JanLahmann/Fun-with-Quantum/master?filepath=GHZ-on-Real-Devices.ipynb) 6. Hardy's Paradox (under development) [![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/JanLahmann/Fun-with-Quantum/master?filepath=Hardys-Paradox.ipynb) +6. Mermin-Peres Magic Square (under development) [![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/JanLahmann/Fun-with-Quantum/master?filepath=Mermin–Peres-Game.ipynb) --- ### IEEE QCE20 Tutorial on "Serious Games for Quantum Computing" @@ -72,7 +73,13 @@ View - and play the game online on real quantum devices, without any install - i ### 6. Hardy's Paradox (under development) A tutorial that discusses a specific version of the Einstein-Podolsky-Rosen (EPR) Paradox - implemented by Jan-R. Lahmann using [Qiskit](https://qiskit.org), [binder](https://mybinder.org) and [RISE](https://rise.readthedocs.io/en/stable/), based on an idea in a former version of the [Qiskit Textbook](https://qiskit.org/textbook) -View - and play the game online, without any install - in Binder: [![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/JanLahmann/Fun-with-Quantum/master?filepath=Hardys-Paradox.ipynb) +View - and play the game online, without any install - in Binder: [![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/JanLahmann/Fun-with-Quantum/master?filepath=Hardys-Paradox.ipynb) + +--- +### 8. Mermin-Peres Magic Square (under development) +A base version of the Mermin-Peres quantum game demonstrating the computational power of quantum computers, which can be further extended to outperform classical machines - extended by Jan-R. Lahmann & David Drexlin with [Qiskit](https://qiskit.org). Based on the Medium article [This Proof Demonstrates a Quantum Advantage, Even for Noisy Quantum Computers](https://medium.com/qiskit/this-proof-demonstrates-a-quantum-advantage-even-for-noisy-quantum-computers-b44a738801ad). + +View - and play the game online, without any install - in Binder: [![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/JanLahmann/Fun-with-Quantum/master?filepath=Mermin–Peres-Game.ipynb) --- ## Usage instructions for the RISE Slideshow Extension @@ -89,3 +96,4 @@ Navigation is easy: --- Jan-R. Lahmann, http://twitter.com/JanLahmann + From 1cfd408a77c61fd8db96f3775cdb8870d133d97b Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Tue, 9 Mar 2021 16:05:15 +0100 Subject: [PATCH 02/30] fix: mybinder-env, more output information --- "Mermin\342\200\223Peres-Game.ipynb" | 84 ++++++++++++++-------------- environment.yml | 1 + 2 files changed, 43 insertions(+), 42 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index f6dc571..a216b1f 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -46,7 +46,6 @@ "outputs": [], "source": [ "def check_result(results, col, row):\n", - "\n", " for i in results: \n", " i = np.array(list(i.split())).astype(np.int)[::-1]\n", " # bring array in correct i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", @@ -82,6 +81,48 @@ " " ] }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "#The following function sets up the initial entanglement between Alice and Bob\n", + "def share_bell_state(qc,a,b,c,d): \n", + " qc.h(a)\n", + " qc.h(b)\n", + " qc.cx(a,c)\n", + " qc.cx(b,d)\n", + " \n", + "# The following functions represent the U(gamma) and V(gamma) controlled Cliffords.\n", + "def U(qc,gamma,a,b):\n", + " if gamma==1:\n", + " qc.h(a)\n", + " qc.i(b)\n", + " elif gamma==2:\n", + " qc.swap(a,b)\n", + " qc.h(a)\n", + " qc.i(b)\n", + "\n", + " elif gamma==3:\n", + " qc.cx(a,b)\n", + " qc.h(a)\n", + " qc.i(b)\n", + "\n", + "def V(qc,gamma,a,b):\n", + " if gamma==1:\n", + " qc.h(a)\n", + " qc.h(b)\n", + " if gamma==2:\n", + " qc.swap(a,b)\n", + " elif gamma==3:\n", + " qc.z(a)\n", + " qc.z(b)\n", + " qc.cz(a,b)\n", + " qc.h(a)\n", + " qc.h(b)" + ] + }, { "cell_type": "code", "execution_count": 5, @@ -171,45 +212,6 @@ } ], "source": [ - "#The following function sets up the initial entanglement between Alice and Bob\n", - "def share_bell_state(qc,a,b,c,d): \n", - " qc.h(a)\n", - " qc.h(b)\n", - " qc.cx(a,c)\n", - " qc.cx(b,d)\n", - " \n", - "# The following functions represent the U(gamma) and V(gamma) controlled Cliffords.\n", - "def U(qc,gamma,a,b):\n", - " if gamma==1:\n", - " qc.h(a)\n", - " qc.i(b)\n", - " elif gamma==2:\n", - " qc.swap(a,b)#\n", - " qc.h(a)\n", - " qc.i(b)\n", - "# qc.swap(a,b)\n", - " elif gamma==3:\n", - " qc.cx(a,b)#\n", - " qc.h(a)\n", - " qc.i(b)\n", - "# qc.cx(a,b)\n", - "def V(qc,gamma,a,b):\n", - " if gamma==1:\n", - " qc.h(a)\n", - " qc.h(b)\n", - " if gamma==2:\n", - " qc.swap(a,b)\n", - " elif gamma==3:\n", - " qc.z(a)#\n", - " qc.z(b)#\n", - " qc.cz(a,b)#\n", - " qc.h(a)\n", - " qc.h(b)\n", - "# qc.cz(a,b)\n", - "# qc.z(a)\n", - "# qc.z(b)\n", - "\n", - "\n", "#We will ask the user to select the row and column on Alice and Bob's behalf.\n", "print(\"Alice and Bob, select a column or row value of 1, 2, or 3\") \n", "print(\"Alice choice (column):\")\n", @@ -250,8 +252,6 @@ " results = job.result().get_counts()\n", " check_result(results, alpha, beta)\n", "\n", - " \n", - " \n", "else:\n", " print(\"please enter values of 1, 2, or 3\")" ] diff --git a/environment.yml b/environment.yml index 31ec25c..da862de 100644 --- a/environment.yml +++ b/environment.yml @@ -11,3 +11,4 @@ dependencies: - tabulate - pylatexenc - seaborn + - numpy From 84666013efc37e65cf78251c216741d1db40e647 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Tue, 9 Mar 2021 16:38:13 +0100 Subject: [PATCH 03/30] fix: print statments --- "Mermin\342\200\223Peres-Game.ipynb" | 194 +++++++++++++++++++++------ 1 file changed, 150 insertions(+), 44 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index a216b1f..58522b9 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -10,7 +10,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.1-final" + "version": "3.8.5-final" }, "orig_nbformat": 2, "kernelspec": { @@ -41,15 +41,16 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 13, "metadata": {}, "outputs": [], "source": [ "def check_result(results, col, row):\n", " for i in results: \n", + " print(\"Quantum circuit solution\")\n", + " print(i+\"\\n\")\n", " i = np.array(list(i.split())).astype(np.int)[::-1]\n", " # bring array in correct i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", - " global a\n", " a = i[0:2]\n", " a_sum = np.sum(a)\n", " a3 = int(np.mod(a_sum+1,2))\n", @@ -66,6 +67,9 @@ "\n", " # compute overall winning condition -> alice at index of bob's row and vice versa\n", " if(winning_bob & winning_alice & (a[row-1]==b[col-1])):\n", + " print(\"Alice's selection\"+\"\\n\"+str(a.reshape(3,1))+\"\\n\")\n", + " print(\"Quantum circuit solution\"+\"\\n\"+str(b)+\"\\n\")\n", + "\n", " print(\"Victory!\") \n", " print(\"The following magic square wins the game!\")\n", " k = np.empty((3,3,))\n", @@ -73,17 +77,16 @@ " k[:,col-1] = a\n", " k[row-1,:] = b\n", " print(k)\n", - "\n", + " print(\"------------------------------------------\") \n", " else:\n", " \"The Game is lost!\"\n", - "\n", " print(\"\\n\")\n", " " ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 14, "metadata": {}, "outputs": [], "source": [ @@ -125,7 +128,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 15, "metadata": {}, "outputs": [ { @@ -135,77 +138,173 @@ "Alice and Bob, select a column or row value of 1, 2, or 3\n", "Alice choice (column):\n", "Bob choice (row):\n", - " ┌───┐ ░ \n", - "q7_0: ┤ H ├──■────────░─\n", - " ├───┤ │ ░ \n", - "q7_1: ┤ H ├──┼────■───░─\n", - " └───┘┌─┴─┐ │ ░ \n", - "q8_0: ─────┤ X ├──┼───░─\n", - " └───┘┌─┴─┐ ░ \n", - "q8_1: ──────────┤ X ├─░─\n", - " └───┘ ░ \n", - "c4: 1/══════════════════\n", - " \n", - "c5: 1/══════════════════\n", - " \n", - "c6: 1/══════════════════\n", - " \n", - "c7: 1/══════════════════\n", - " \n", + " ┌───┐ ░ \n", + "q21_0: ┤ H ├──■────────░─\n", + " ├───┤ │ ░ \n", + "q21_1: ┤ H ├──┼────■───░─\n", + " └───┘┌─┴─┐ │ ░ \n", + "q22_0: ─────┤ X ├──┼───░─\n", + " └───┘┌─┴─┐ ░ \n", + "q22_1: ──────────┤ X ├─░─\n", + " └───┘ ░ \n", + "c12: 1/══════════════════\n", + " \n", + "c13: 1/══════════════════\n", + " \n", + "c14: 1/══════════════════\n", + " \n", + "c15: 1/══════════════════\n", + " \n", + "Quantum circuit solution\n", + "1 0 1 0\n", + "\n", + "Alice's selection\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Quantum circuit solution\n", + "[0 1 1]\n", + "\n", "Victory!\n", - "The following is the winning magic square!\n", - "[[ 1. 1. 0.]\n", + "The following magic square wins the game!\n", + "[[ 0. 1. 1.]\n", " [ 1. nan nan]\n", - " [ 1. nan nan]]\n", + " [ 0. nan nan]]\n", + "------------------------------------------\n", + "\n", "\n", + "Quantum circuit solution\n", + "0 0 0 0\n", + "\n", + "Alice's selection\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Quantum circuit solution\n", + "[0 0 0]\n", "\n", "Victory!\n", - "The following is the winning magic square!\n", - "[[ 1. 1. 0.]\n", + "The following magic square wins the game!\n", + "[[ 0. 0. 0.]\n", " [ 0. nan nan]\n", - " [ 0. nan nan]]\n", + " [ 1. nan nan]]\n", + "------------------------------------------\n", + "\n", "\n", + "Quantum circuit solution\n", + "0 1 1 1\n", + "\n", + "Alice's selection\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Quantum circuit solution\n", + "[1 0 1]\n", "\n", "Victory!\n", - "The following is the winning magic square!\n", + "The following magic square wins the game!\n", "[[ 1. 0. 1.]\n", " [ 1. nan nan]\n", " [ 1. nan nan]]\n", + "------------------------------------------\n", + "\n", "\n", + "Quantum circuit solution\n", + "0 0 1 0\n", + "\n", + "Alice's selection\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Quantum circuit solution\n", + "[0 0 0]\n", "\n", "Victory!\n", - "The following is the winning magic square!\n", - "[[ 0. 1. 1.]\n", + "The following magic square wins the game!\n", + "[[ 0. 0. 0.]\n", " [ 1. nan nan]\n", " [ 0. nan nan]]\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1 1 1 1\n", + "\n", + "Alice's selection\n", + "[[1]\n", + " [1]\n", + " [1]]\n", "\n", + "Quantum circuit solution\n", + "[1 1 0]\n", "\n", "Victory!\n", - "The following is the winning magic square!\n", + "The following magic square wins the game!\n", + "[[ 1. 1. 0.]\n", + " [ 1. nan nan]\n", + " [ 1. nan nan]]\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0 1 0 1\n", + "\n", + "Alice's selection\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Quantum circuit solution\n", + "[1 0 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", "[[ 1. 0. 1.]\n", " [ 0. nan nan]\n", " [ 0. nan nan]]\n", + "------------------------------------------\n", "\n", "\n", - "Victory!\n", - "The following is the winning magic square!\n", - "[[ 0. 0. 0.]\n", - " [ 1. nan nan]\n", - " [ 0. nan nan]]\n", + "Quantum circuit solution\n", + "1 1 0 1\n", + "\n", + "Alice's selection\n", + "[[1]\n", + " [0]\n", + " [0]]\n", "\n", + "Quantum circuit solution\n", + "[1 1 0]\n", "\n", "Victory!\n", - "The following is the winning magic square!\n", - "[[ 0. 1. 1.]\n", + "The following magic square wins the game!\n", + "[[ 1. 1. 0.]\n", " [ 0. nan nan]\n", - " [ 1. nan nan]]\n", + " [ 0. nan nan]]\n", + "------------------------------------------\n", + "\n", "\n", + "Quantum circuit solution\n", + "1 0 0 0\n", + "\n", + "Alice's selection\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Quantum circuit solution\n", + "[0 1 1]\n", "\n", "Victory!\n", - "The following is the winning magic square!\n", - "[[ 0. 0. 0.]\n", + "The following magic square wins the game!\n", + "[[ 0. 1. 1.]\n", " [ 0. nan nan]\n", " [ 1. nan nan]]\n", + "------------------------------------------\n", "\n", "\n" ] @@ -255,6 +354,13 @@ "else:\n", " print(\"please enter values of 1, 2, or 3\")" ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] } ] } \ No newline at end of file From 0cedae40c6f32a346e972e4208939fd4c23da904 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Tue, 9 Mar 2021 17:47:10 +0100 Subject: [PATCH 04/30] fix: deprecated numpy usage --- "Mermin\342\200\223Peres-Game.ipynb" | 172 ++++++++++++++------------- 1 file changed, 92 insertions(+), 80 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 58522b9..038a27b 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -31,25 +31,28 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 52, "metadata": {}, "outputs": [], "source": [ "from qiskit import *\n", - "import numpy as np" + "import numpy as np\n", + "from qiskit.test.mock import *" ] }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 63, "metadata": {}, "outputs": [], "source": [ "def check_result(results, col, row):\n", + " lost = 0 \n", + " won = 0\n", " for i in results: \n", " print(\"Quantum circuit solution\")\n", " print(i+\"\\n\")\n", - " i = np.array(list(i.split())).astype(np.int)[::-1]\n", + " i = np.array(list(i.split())).astype(int)[::-1]\n", " # bring array in correct i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", " a = i[0:2]\n", " a_sum = np.sum(a)\n", @@ -66,10 +69,10 @@ " # compute y3 by y1+y2 mod 2 and save to array\n", "\n", " # compute overall winning condition -> alice at index of bob's row and vice versa\n", + " print(\"Alice's selection. Column:\"+str(col)+\"\\n\"+str(a.reshape(3,1))+\"\\n\")\n", + " print(\"Bob's circuit solution. Row:\"+str(row)+\"\\n\"+str(b)+\"\\n\")\n", + " \n", " if(winning_bob & winning_alice & (a[row-1]==b[col-1])):\n", - " print(\"Alice's selection\"+\"\\n\"+str(a.reshape(3,1))+\"\\n\")\n", - " print(\"Quantum circuit solution\"+\"\\n\"+str(b)+\"\\n\")\n", - "\n", " print(\"Victory!\") \n", " print(\"The following magic square wins the game!\")\n", " k = np.empty((3,3,))\n", @@ -77,16 +80,20 @@ " k[:,col-1] = a\n", " k[row-1,:] = b\n", " print(k)\n", - " print(\"------------------------------------------\") \n", + " won += \n", " else:\n", - " \"The Game is lost!\"\n", + " print(\"The Game is lost!\")\n", + " lost += \n", + " print(\"------------------------------------------\") \n", " print(\"\\n\")\n", - " " + " print(lost)\n", + " print(won)\n", + " print(\"Accuracy:\"+str(won/(lost+won)))" ] }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 64, "metadata": {}, "outputs": [], "source": [ @@ -128,7 +135,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 66, "metadata": {}, "outputs": [ { @@ -138,113 +145,113 @@ "Alice and Bob, select a column or row value of 1, 2, or 3\n", "Alice choice (column):\n", "Bob choice (row):\n", - " ┌───┐ ░ \n", - "q21_0: ┤ H ├──■────────░─\n", - " ├───┤ │ ░ \n", - "q21_1: ┤ H ├──┼────■───░─\n", - " └───┘┌─┴─┐ │ ░ \n", - "q22_0: ─────┤ X ├──┼───░─\n", - " └───┘┌─┴─┐ ░ \n", - "q22_1: ──────────┤ X ├─░─\n", - " └───┘ ░ \n", - "c12: 1/══════════════════\n", - " \n", - "c13: 1/══════════════════\n", - " \n", - "c14: 1/══════════════════\n", - " \n", - "c15: 1/══════════════════\n", - " \n", + " ┌───┐ ░ \n", + "q387_0: ┤ H ├──■────────░─\n", + " ├───┤ │ ░ \n", + "q387_1: ┤ H ├──┼────■───░─\n", + " └───┘┌─┴─┐ │ ░ \n", + "q388_0: ─────┤ X ├──┼───░─\n", + " └───┘┌─┴─┐ ░ \n", + "q388_1: ──────────┤ X ├─░─\n", + " └───┘ ░ \n", + "c100: 1/══════════════════\n", + " \n", + "c101: 1/══════════════════\n", + " \n", + "c102: 1/══════════════════\n", + " \n", + "c103: 1/══════════════════\n", + " \n", "Quantum circuit solution\n", - "1 0 1 0\n", + "0 0 1 0\n", "\n", - "Alice's selection\n", + "Alice's selection. Column:1\n", "[[0]\n", " [1]\n", " [0]]\n", "\n", - "Quantum circuit solution\n", - "[0 1 1]\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", - "[[ 0. 1. 1.]\n", + "[[ 0. 0. 0.]\n", " [ 1. nan nan]\n", " [ 0. nan nan]]\n", "------------------------------------------\n", "\n", "\n", "Quantum circuit solution\n", - "0 0 0 0\n", + "1 0 0 0\n", "\n", - "Alice's selection\n", + "Alice's selection. Column:1\n", "[[0]\n", " [0]\n", " [1]]\n", "\n", - "Quantum circuit solution\n", - "[0 0 0]\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", - "[[ 0. 0. 0.]\n", + "[[ 0. 1. 1.]\n", " [ 0. nan nan]\n", " [ 1. nan nan]]\n", "------------------------------------------\n", "\n", "\n", "Quantum circuit solution\n", - "0 1 1 1\n", + "1 0 1 0\n", "\n", - "Alice's selection\n", - "[[1]\n", + "Alice's selection. Column:1\n", + "[[0]\n", " [1]\n", - " [1]]\n", + " [0]]\n", "\n", - "Quantum circuit solution\n", - "[1 0 1]\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", - "[[ 1. 0. 1.]\n", + "[[ 0. 1. 1.]\n", " [ 1. nan nan]\n", - " [ 1. nan nan]]\n", + " [ 0. nan nan]]\n", "------------------------------------------\n", "\n", "\n", "Quantum circuit solution\n", - "0 0 1 0\n", + "1 1 1 1\n", "\n", - "Alice's selection\n", - "[[0]\n", + "Alice's selection. Column:1\n", + "[[1]\n", " [1]\n", - " [0]]\n", + " [1]]\n", "\n", - "Quantum circuit solution\n", - "[0 0 0]\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", - "[[ 0. 0. 0.]\n", + "[[ 1. 1. 0.]\n", " [ 1. nan nan]\n", - " [ 0. nan nan]]\n", + " [ 1. nan nan]]\n", "------------------------------------------\n", "\n", "\n", "Quantum circuit solution\n", - "1 1 1 1\n", + "0 1 1 1\n", "\n", - "Alice's selection\n", + "Alice's selection. Column:1\n", "[[1]\n", " [1]\n", " [1]]\n", "\n", - "Quantum circuit solution\n", - "[1 1 0]\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", - "[[ 1. 1. 0.]\n", + "[[ 1. 0. 1.]\n", " [ 1. nan nan]\n", " [ 1. nan nan]]\n", "------------------------------------------\n", @@ -253,12 +260,12 @@ "Quantum circuit solution\n", "0 1 0 1\n", "\n", - "Alice's selection\n", + "Alice's selection. Column:1\n", "[[1]\n", " [0]\n", " [0]]\n", "\n", - "Quantum circuit solution\n", + "Bob's circuit solution. Row:1\n", "[1 0 1]\n", "\n", "Victory!\n", @@ -270,43 +277,47 @@ "\n", "\n", "Quantum circuit solution\n", - "1 1 0 1\n", + "0 0 0 0\n", "\n", - "Alice's selection\n", - "[[1]\n", + "Alice's selection. Column:1\n", + "[[0]\n", " [0]\n", - " [0]]\n", + " [1]]\n", "\n", - "Quantum circuit solution\n", - "[1 1 0]\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", - "[[ 1. 1. 0.]\n", + "[[ 0. 0. 0.]\n", " [ 0. nan nan]\n", - " [ 0. nan nan]]\n", + " [ 1. nan nan]]\n", "------------------------------------------\n", "\n", "\n", "Quantum circuit solution\n", - "1 0 0 0\n", + "1 1 0 1\n", "\n", - "Alice's selection\n", - "[[0]\n", + "Alice's selection. Column:1\n", + "[[1]\n", " [0]\n", - " [1]]\n", + " [0]]\n", "\n", - "Quantum circuit solution\n", - "[0 1 1]\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", - "[[ 0. 1. 1.]\n", + "[[ 1. 1. 0.]\n", " [ 0. nan nan]\n", - " [ 1. nan nan]]\n", + " [ 0. nan nan]]\n", "------------------------------------------\n", "\n", - "\n" + "\n", + "0\n", + "8\n", + "Accuracy:1.0\n", + "{'0 0 1 0': 1062, '1 0 0 0': 1022, '1 0 1 0': 1015, '1 1 1 1': 1009, '0 1 1 1': 1018, '0 1 0 1': 1036, '0 0 0 0': 1049, '1 1 0 1': 981}\n" ] } ], @@ -347,9 +358,10 @@ " magicsquare_circuit.measure(3,3)\n", " \n", " backend = BasicAer.get_backend('qasm_simulator') # define the backend\n", - " job = execute(magicsquare_circuit, backend, shots=100) # run the job simulation\n", + " job = execute(magicsquare_circuit, backend, shots=8192, optimization_level=3) # run the job simulation\n", " results = job.result().get_counts()\n", " check_result(results, alpha, beta)\n", + " print(results)\n", "\n", "else:\n", " print(\"please enter values of 1, 2, or 3\")" From 4cf69ce6d198697107337f329048c8a03c1a0bb4 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Wed, 10 Mar 2021 09:03:56 +0100 Subject: [PATCH 05/30] fix: env --- environment.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/environment.yml b/environment.yml index da862de..f62a03a 100644 --- a/environment.yml +++ b/environment.yml @@ -6,9 +6,9 @@ dependencies: - ipywidgets - rise - pip: + - numpy==1.19.5 - qiskit==0.23 - wget - tabulate - pylatexenc - seaborn - - numpy From 00ed7106861a89164d74292da8a5d063a965c6c4 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Wed, 10 Mar 2021 10:53:56 +0100 Subject: [PATCH 06/30] fix: print explaination --- "Mermin\342\200\223Peres-Game.ipynb" | 213 +++++++++++++++++++-------- 1 file changed, 148 insertions(+), 65 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 038a27b..02bda2f 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -31,25 +31,29 @@ }, { "cell_type": "code", - "execution_count": 52, + "execution_count": 109, "metadata": {}, "outputs": [], "source": [ - "from qiskit import *\n", + "from qiskit import *#\n", "import numpy as np\n", "from qiskit.test.mock import *" ] }, { "cell_type": "code", - "execution_count": 63, + "execution_count": 110, "metadata": {}, "outputs": [], "source": [ "def check_result(results, col, row):\n", - " lost = 0 \n", - " won = 0\n", + " print_tut = True\n", " for i in results: \n", + " \n", + " if print_tut:\n", + " explain(i, col, row)\n", + " print_tut = False\n", + "\n", " print(\"Quantum circuit solution\")\n", " print(i+\"\\n\")\n", " i = np.array(list(i.split())).astype(int)[::-1]\n", @@ -80,20 +84,70 @@ " k[:,col-1] = a\n", " k[row-1,:] = b\n", " print(k)\n", - " won += \n", " else:\n", " print(\"The Game is lost!\")\n", - " lost += \n", " print(\"------------------------------------------\") \n", - " print(\"\\n\")\n", - " print(lost)\n", - " print(won)\n", - " print(\"Accuracy:\"+str(won/(lost+won)))" + " print(\"\\n\")\n" + ] + }, + { + "cell_type": "code", + "execution_count": 123, + "metadata": {}, + "outputs": [], + "source": [ + "def explain(result, col, row):\n", + " print(\"Resultant quantum circuit solution:\"+\"\\n\")\n", + " print(result+\"\\n\")\n", + " example = np.array(list(result.split())).astype(int)[::-1]\n", + " print(\"Bring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\"+\"\\n\")\n", + " print(str(example)+\"\\n\")\n", + " print(\"Compute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\")\n", + " print(\"Compute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\"+\"\\n\")\n", + " \n", + " a = example[0:2]\n", + " a_sum = np.sum(a)\n", + " a3 = int(np.mod(a_sum+1,2))\n", + " a = np.append(a,a3)\n", + " winning_alice = np.mod(a_sum + a3, 2) == 1\n", + "\n", + " b = example[2:]\n", + " b_sum = np.sum(b)\n", + " b3 = int(np.mod(b_sum,2))\n", + " b = np.append(b,b3)\n", + " winning_bob = np.mod(b_sum + b3, 2) == 0 \n", + "\n", + " print(\"Alice's column solution\"+\"\\n\"+str(a.reshape(3,1))+\"\\n\" + \"Winning conditions fullfilled (Odd quantitude of 1)? :\"+str(winning_alice)+\"\\n\")\n", + " print(\"Bob's row solution\"+\"\\n\"+str(b)+\"\\n\"+ \"Winning conditions fullfilled (Even quantitude of 1)? :\"+str(winning_bob)+\"\\n\")\n", + " print(\"Create matrix to show results\"+\"\\n\")\n", + " k = np.empty((3,3,))\n", + " k[:] = np.nan\n", + " k[:,col-1] = a\n", + " k[row-1,:] = b\n", + " print(k)\n", + " print(\"------------------------------------------\") \n", + " print(\"------------------------------------------\") " + ] + }, + { + "cell_type": "code", + "execution_count": 124, + "metadata": {}, + "outputs": [], + "source": [ + "\n", + "def get_state_accuracy(counts): # funciton to calculate state accuracy\n", + " expected_counts = 0\n", + " for state in counts.keys():\n", + " if state in expected_states:\n", + " expected_counts = expected_counts + counts[state]\n", + " state_accuracy = expected_counts / shots\n", + " return str(state_accuracy*100)+\"%\"" ] }, { "cell_type": "code", - "execution_count": 64, + "execution_count": 125, "metadata": {}, "outputs": [], "source": [ @@ -135,7 +189,7 @@ }, { "cell_type": "code", - "execution_count": 66, + "execution_count": 126, "metadata": {}, "outputs": [ { @@ -145,68 +199,61 @@ "Alice and Bob, select a column or row value of 1, 2, or 3\n", "Alice choice (column):\n", "Bob choice (row):\n", - " ┌───┐ ░ \n", - "q387_0: ┤ H ├──■────────░─\n", - " ├───┤ │ ░ \n", - "q387_1: ┤ H ├──┼────■───░─\n", - " └───┘┌─┴─┐ │ ░ \n", - "q388_0: ─────┤ X ├──┼───░─\n", - " └───┘┌─┴─┐ ░ \n", - "q388_1: ──────────┤ X ├─░─\n", - " └───┘ ░ \n", - "c100: 1/══════════════════\n", - " \n", - "c101: 1/══════════════════\n", - " \n", - "c102: 1/══════════════════\n", - " \n", - "c103: 1/══════════════════\n", - " \n", - "Quantum circuit solution\n", - "0 0 1 0\n", + " ┌───┐ ░ ┌───┐┌─┐ ┌─┐\n", + "q527_0: ┤ H ├──■────────░─┤ H ├┤M├─────────┤M├\n", + " ├───┤ │ ░ ├───┤└╥┘┌─┐ └╥┘\n", + "q527_1: ┤ H ├──┼────■───░─┤ I ├─╫─┤M├───────╫─\n", + " └───┘┌─┴─┐ │ ░ ├───┤ ║ └╥┘┌─┐ ║ \n", + "q528_0: ─────┤ X ├──┼───░─┤ H ├─╫──╫─┤M├────╫─\n", + " └───┘┌─┴─┐ ░ ├───┤ ║ ║ └╥┘┌─┐ ║ \n", + "q528_1: ──────────┤ X ├─░─┤ H ├─╫──╫──╫─┤M├─╫─\n", + " └───┘ ░ └───┘ ║ ║ ║ └╥┘ ║ \n", + "c156: 1/════════════════════════╩══╬══╬══╬══╩═\n", + " 0 ║ ║ ║ 0 \n", + " ║ ║ ║ \n", + "c157: 1/═══════════════════════════╩══╬══╬════\n", + " 0 ║ ║ \n", + " ║ ║ \n", + "c158: 1/══════════════════════════════╩══╬════\n", + " 0 ║ \n", + " ║ \n", + "c159: 1/═════════════════════════════════╩════\n", + " 0 \n", + "Resultant quantum circuit solution:\n", "\n", - "Alice's selection. Column:1\n", - "[[0]\n", - " [1]\n", - " [0]]\n", + "1 0 0 0\n", "\n", - "Bob's circuit solution. Row:1\n", - "[0 0 0]\n", + "Bring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "[[ 0. 0. 0.]\n", - " [ 1. nan nan]\n", - " [ 0. nan nan]]\n", - "------------------------------------------\n", + "[0 0 0 1]\n", "\n", + "Compute missing x3 with x1+x2+1 mod 2 and save to array for further processing\n", + "Compute missing y3 with y1+y2 mod 2 and save to array for further processing\n", "\n", - "Quantum circuit solution\n", - "1 0 0 0\n", - "\n", - "Alice's selection. Column:1\n", + "Alice's column solution\n", "[[0]\n", " [0]\n", " [1]]\n", + "Winning conditions fullfilled (Odd quantitude of 1)? :True\n", "\n", - "Bob's circuit solution. Row:1\n", + "Bob's row solution\n", "[0 1 1]\n", + "Winning conditions fullfilled (Even quantitude of 1)? :True\n", + "\n", + "Create matrix to show results\n", "\n", - "Victory!\n", - "The following magic square wins the game!\n", "[[ 0. 1. 1.]\n", " [ 0. nan nan]\n", " [ 1. nan nan]]\n", "------------------------------------------\n", - "\n", - "\n", + "------------------------------------------\n", "Quantum circuit solution\n", - "1 0 1 0\n", + "1 0 0 0\n", "\n", "Alice's selection. Column:1\n", "[[0]\n", - " [1]\n", - " [0]]\n", + " [0]\n", + " [1]]\n", "\n", "Bob's circuit solution. Row:1\n", "[0 1 1]\n", @@ -214,8 +261,8 @@ "Victory!\n", "The following magic square wins the game!\n", "[[ 0. 1. 1.]\n", - " [ 1. nan nan]\n", - " [ 0. nan nan]]\n", + " [ 0. nan nan]\n", + " [ 1. nan nan]]\n", "------------------------------------------\n", "\n", "\n", @@ -258,6 +305,25 @@ "\n", "\n", "Quantum circuit solution\n", + "1 0 1 0\n", + "\n", + "Alice's selection. Column:1\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "[[ 0. 1. 1.]\n", + " [ 1. nan nan]\n", + " [ 0. nan nan]]\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", "0 1 0 1\n", "\n", "Alice's selection. Column:1\n", @@ -296,6 +362,25 @@ "\n", "\n", "Quantum circuit solution\n", + "0 0 1 0\n", + "\n", + "Alice's selection. Column:1\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "[[ 0. 0. 0.]\n", + " [ 1. nan nan]\n", + " [ 0. nan nan]]\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", "1 1 0 1\n", "\n", "Alice's selection. Column:1\n", @@ -314,10 +399,7 @@ "------------------------------------------\n", "\n", "\n", - "0\n", - "8\n", - "Accuracy:1.0\n", - "{'0 0 1 0': 1062, '1 0 0 0': 1022, '1 0 1 0': 1015, '1 1 1 1': 1009, '0 1 1 1': 1018, '0 1 0 1': 1036, '0 0 0 0': 1049, '1 1 0 1': 981}\n" + "{'1 0 0 0': 998, '1 1 1 1': 1052, '0 1 1 1': 980, '1 0 1 0': 1037, '0 1 0 1': 1055, '0 0 0 0': 1037, '0 0 1 0': 1024, '1 1 0 1': 1009}\n" ] } ], @@ -344,8 +426,6 @@ "share_bell_state(magicsquare_circuit,0,1,2,3)\n", "\n", "magicsquare_circuit.barrier()\n", - "magicsquare_circuit.draw(output='mpl')\n", - "print(magicsquare_circuit)\n", "\n", "#Draw the rest of the circuit based on Alice and Bob's selection \n", "if 4>alpha>0 and 4>beta>0:\n", @@ -357,8 +437,11 @@ " magicsquare_circuit.measure(2,2)\n", " magicsquare_circuit.measure(3,3)\n", " \n", + " magicsquare_circuit.draw(output='mpl')\n", + " print(magicsquare_circuit)\n", + "\n", " backend = BasicAer.get_backend('qasm_simulator') # define the backend\n", - " job = execute(magicsquare_circuit, backend, shots=8192, optimization_level=3) # run the job simulation\n", + " job = execute(magicsquare_circuit, backend, shots=8192) # run the job simulation\n", " results = job.result().get_counts()\n", " check_result(results, alpha, beta)\n", " print(results)\n", From 4dd97a8898b00bbc0da8a68ee2f049e70c35fff4 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Wed, 10 Mar 2021 12:41:07 +0100 Subject: [PATCH 07/30] fix: circuit code duplication --- "Mermin\342\200\223Peres-Game.ipynb" | 122 +++++++++++++-------------- 1 file changed, 61 insertions(+), 61 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 02bda2f..21a0f0f 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -189,7 +189,7 @@ }, { "cell_type": "code", - "execution_count": 126, + "execution_count": 127, "metadata": {}, "outputs": [ { @@ -199,75 +199,75 @@ "Alice and Bob, select a column or row value of 1, 2, or 3\n", "Alice choice (column):\n", "Bob choice (row):\n", - " ┌───┐ ░ ┌───┐┌─┐ ┌─┐\n", - "q527_0: ┤ H ├──■────────░─┤ H ├┤M├─────────┤M├\n", - " ├───┤ │ ░ ├───┤└╥┘┌─┐ └╥┘\n", - "q527_1: ┤ H ├──┼────■───░─┤ I ├─╫─┤M├───────╫─\n", - " └───┘┌─┴─┐ │ ░ ├───┤ ║ └╥┘┌─┐ ║ \n", - "q528_0: ─────┤ X ├──┼───░─┤ H ├─╫──╫─┤M├────╫─\n", - " └───┘┌─┴─┐ ░ ├───┤ ║ ║ └╥┘┌─┐ ║ \n", - "q528_1: ──────────┤ X ├─░─┤ H ├─╫──╫──╫─┤M├─╫─\n", - " └───┘ ░ └───┘ ║ ║ ║ └╥┘ ║ \n", - "c156: 1/════════════════════════╩══╬══╬══╬══╩═\n", - " 0 ║ ║ ║ 0 \n", - " ║ ║ ║ \n", - "c157: 1/═══════════════════════════╩══╬══╬════\n", - " 0 ║ ║ \n", - " ║ ║ \n", - "c158: 1/══════════════════════════════╩══╬════\n", - " 0 ║ \n", - " ║ \n", - "c159: 1/═════════════════════════════════╩════\n", - " 0 \n", + " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", + "q534_0: ┤ H ├──■────────░─┤ H ├─░─┤M├─────────\n", + " ├───┤ │ ░ ├───┤ ░ └╥┘┌─┐ \n", + "q534_1: ┤ H ├──┼────■───░─┤ I ├─░──╫─┤M├──────\n", + " └───┘┌─┴─┐ │ ░ ├───┤ ░ ║ └╥┘┌─┐ \n", + "q535_0: ─────┤ X ├──┼───░─┤ H ├─░──╫──╫─┤M├───\n", + " └───┘┌─┴─┐ ░ ├───┤ ░ ║ ║ └╥┘┌─┐\n", + "q535_1: ──────────┤ X ├─░─┤ H ├─░──╫──╫──╫─┤M├\n", + " └───┘ ░ └───┘ ░ ║ ║ ║ └╥┘\n", + "c160: 1/═══════════════════════════╩══╬══╬══╬═\n", + " 0 ║ ║ ║ \n", + " ║ ║ ║ \n", + "c161: 1/══════════════════════════════╩══╬══╬═\n", + " 0 ║ ║ \n", + " ║ ║ \n", + "c162: 1/═════════════════════════════════╩══╬═\n", + " 0 ║ \n", + " ║ \n", + "c163: 1/════════════════════════════════════╩═\n", + " 0 \n", "Resultant quantum circuit solution:\n", "\n", - "1 0 0 0\n", + "0 0 1 0\n", "\n", "Bring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", "\n", - "[0 0 0 1]\n", + "[0 1 0 0]\n", "\n", "Compute missing x3 with x1+x2+1 mod 2 and save to array for further processing\n", "Compute missing y3 with y1+y2 mod 2 and save to array for further processing\n", "\n", "Alice's column solution\n", "[[0]\n", - " [0]\n", - " [1]]\n", + " [1]\n", + " [0]]\n", "Winning conditions fullfilled (Odd quantitude of 1)? :True\n", "\n", "Bob's row solution\n", - "[0 1 1]\n", + "[0 0 0]\n", "Winning conditions fullfilled (Even quantitude of 1)? :True\n", "\n", "Create matrix to show results\n", "\n", - "[[ 0. 1. 1.]\n", - " [ 0. nan nan]\n", - " [ 1. nan nan]]\n", + "[[ 0. 0. 0.]\n", + " [ 1. nan nan]\n", + " [ 0. nan nan]]\n", "------------------------------------------\n", "------------------------------------------\n", "Quantum circuit solution\n", - "1 0 0 0\n", + "0 0 1 0\n", "\n", "Alice's selection. Column:1\n", "[[0]\n", - " [0]\n", - " [1]]\n", + " [1]\n", + " [0]]\n", "\n", "Bob's circuit solution. Row:1\n", - "[0 1 1]\n", + "[0 0 0]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", - "[[ 0. 1. 1.]\n", - " [ 0. nan nan]\n", - " [ 1. nan nan]]\n", + "[[ 0. 0. 0.]\n", + " [ 1. nan nan]\n", + " [ 0. nan nan]]\n", "------------------------------------------\n", "\n", "\n", "Quantum circuit solution\n", - "1 1 1 1\n", + "0 1 1 1\n", "\n", "Alice's selection. Column:1\n", "[[1]\n", @@ -275,23 +275,23 @@ " [1]]\n", "\n", "Bob's circuit solution. Row:1\n", - "[1 1 0]\n", + "[1 0 1]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", - "[[ 1. 1. 0.]\n", + "[[ 1. 0. 1.]\n", " [ 1. nan nan]\n", " [ 1. nan nan]]\n", "------------------------------------------\n", "\n", "\n", "Quantum circuit solution\n", - "0 1 1 1\n", + "0 1 0 1\n", "\n", "Alice's selection. Column:1\n", "[[1]\n", - " [1]\n", - " [1]]\n", + " [0]\n", + " [0]]\n", "\n", "Bob's circuit solution. Row:1\n", "[1 0 1]\n", @@ -299,8 +299,8 @@ "Victory!\n", "The following magic square wins the game!\n", "[[ 1. 0. 1.]\n", - " [ 1. nan nan]\n", - " [ 1. nan nan]]\n", + " [ 0. nan nan]\n", + " [ 0. nan nan]]\n", "------------------------------------------\n", "\n", "\n", @@ -324,7 +324,7 @@ "\n", "\n", "Quantum circuit solution\n", - "0 1 0 1\n", + "1 1 0 1\n", "\n", "Alice's selection. Column:1\n", "[[1]\n", @@ -332,11 +332,11 @@ " [0]]\n", "\n", "Bob's circuit solution. Row:1\n", - "[1 0 1]\n", + "[1 1 0]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", - "[[ 1. 0. 1.]\n", + "[[ 1. 1. 0.]\n", " [ 0. nan nan]\n", " [ 0. nan nan]]\n", "------------------------------------------\n", @@ -362,44 +362,44 @@ "\n", "\n", "Quantum circuit solution\n", - "0 0 1 0\n", + "1 1 1 1\n", "\n", "Alice's selection. Column:1\n", - "[[0]\n", + "[[1]\n", " [1]\n", - " [0]]\n", + " [1]]\n", "\n", "Bob's circuit solution. Row:1\n", - "[0 0 0]\n", + "[1 1 0]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", - "[[ 0. 0. 0.]\n", + "[[ 1. 1. 0.]\n", " [ 1. nan nan]\n", - " [ 0. nan nan]]\n", + " [ 1. nan nan]]\n", "------------------------------------------\n", "\n", "\n", "Quantum circuit solution\n", - "1 1 0 1\n", + "1 0 0 0\n", "\n", "Alice's selection. Column:1\n", - "[[1]\n", + "[[0]\n", " [0]\n", - " [0]]\n", + " [1]]\n", "\n", "Bob's circuit solution. Row:1\n", - "[1 1 0]\n", + "[0 1 1]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", - "[[ 1. 1. 0.]\n", + "[[ 0. 1. 1.]\n", " [ 0. nan nan]\n", - " [ 0. nan nan]]\n", + " [ 1. nan nan]]\n", "------------------------------------------\n", "\n", "\n", - "{'1 0 0 0': 998, '1 1 1 1': 1052, '0 1 1 1': 980, '1 0 1 0': 1037, '0 1 0 1': 1055, '0 0 0 0': 1037, '0 0 1 0': 1024, '1 1 0 1': 1009}\n" + "{'0 0 1 0': 1041, '0 1 1 1': 1007, '0 1 0 1': 1033, '1 0 1 0': 1014, '1 1 0 1': 1039, '0 0 0 0': 1036, '1 1 1 1': 1028, '1 0 0 0': 994}\n" ] } ], @@ -431,7 +431,7 @@ "if 4>alpha>0 and 4>beta>0:\n", " U(magicsquare_circuit,alpha,0,1) \n", " V(magicsquare_circuit,beta,2,3)\n", - " magicsquare_circuit.measure(0,0)\n", + " magicsquare_circuit.barrier()\n", " magicsquare_circuit.measure(0,0)\n", " magicsquare_circuit.measure(1,1)\n", " magicsquare_circuit.measure(2,2)\n", From ee4ae4d74e391613f0182cc2cdc7a8be88a7c5f6 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Wed, 10 Mar 2021 12:42:06 +0100 Subject: [PATCH 08/30] fix: circuit code duplication --- "Mermin\342\200\223Peres-Game.ipynb" | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 21a0f0f..6c7e4ec 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -117,8 +117,8 @@ " b = np.append(b,b3)\n", " winning_bob = np.mod(b_sum + b3, 2) == 0 \n", "\n", - " print(\"Alice's column solution\"+\"\\n\"+str(a.reshape(3,1))+\"\\n\" + \"Winning conditions fullfilled (Odd quantitude of 1)? :\"+str(winning_alice)+\"\\n\")\n", - " print(\"Bob's row solution\"+\"\\n\"+str(b)+\"\\n\"+ \"Winning conditions fullfilled (Even quantitude of 1)? :\"+str(winning_bob)+\"\\n\")\n", + " print(\"Alice's column solution\"+\"\\n\"+str(a.reshape(3,1))+\"\\n\" + \"Winning conditions fullfilled (Odd quantitude of 1)?: \"+str(winning_alice)+\"\\n\")\n", + " print(\"Bob's row solution\"+\"\\n\"+str(b)+\"\\n\"+ \"Winning conditions fullfilled (Even quantitude of 1)?: \"+str(winning_bob)+\"\\n\")\n", " print(\"Create matrix to show results\"+\"\\n\")\n", " k = np.empty((3,3,))\n", " k[:] = np.nan\n", From b4c2aa90e9d280a4dff1c4aba8b3d36c036dda8b Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Tue, 23 Mar 2021 22:55:23 +0100 Subject: [PATCH 09/30] feat: real devices & math background --- .vscode/settings.json | 3 + "Mermin\342\200\223Peres-Game.ipynb" | 820 +++++++++++++++++++-------- 2 files changed, 588 insertions(+), 235 deletions(-) create mode 100644 .vscode/settings.json diff --git a/.vscode/settings.json b/.vscode/settings.json new file mode 100644 index 0000000..625b81c --- /dev/null +++ b/.vscode/settings.json @@ -0,0 +1,3 @@ +{ + "jupyter.jupyterServerType": "local" +} \ No newline at end of file diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 6c7e4ec..94bde9d 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -10,7 +10,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.5-final" + "version": "3.9.1-final" }, "orig_nbformat": 2, "kernelspec": { @@ -29,33 +29,47 @@ "cell_type": "markdown", "metadata": {} }, + { + "source": [ + "### In order to solve and test the Mirmin Peres Magic Square (MPG) game on a quantum computer, we will employ the Qiskit framework and the IBM Q-Experience. Subsequently, this notebook will introduce the quantum circuit solving the MPG game, quantify it's performance on both simulated and real devices, and finally provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game.\n", + "\n", + "#### For the subsequent circuit definition, evaluation and illustration, we first have to define auxiliary functions and import the corresponding modules from Qiskit and Numpy" + ], + "cell_type": "markdown", + "metadata": {} + }, { "cell_type": "code", - "execution_count": 109, + "execution_count": 365, "metadata": {}, "outputs": [], "source": [ - "from qiskit import *#\n", + "from qiskit import *\n", "import numpy as np\n", - "from qiskit.test.mock import *" + "from qiskit.test.mock import *\n", + "from qiskit.visualization import plot_state_qsphere\n", + "from qiskit_textbook.tools import array_to_latex" ] }, { "cell_type": "code", - "execution_count": 110, + "execution_count": 366, "metadata": {}, "outputs": [], "source": [ + "#checks if the circuit output fulfills the winning criteria\n", "def check_result(results, col, row):\n", " print_tut = True\n", + " expected_states = list()\n", " for i in results: \n", " \n", - " if print_tut:\n", - " explain(i, col, row)\n", - " print_tut = False\n", + " #if print_tut:\n", + " # explain(i, col, row)\n", + " # print_tut = False\n", "\n", - " print(\"Quantum circuit solution\")\n", - " print(i+\"\\n\")\n", + " #print(\"Quantum circuit solution\")\n", + " #print(i+\"\\n\")\n", + " i_orginal = i\n", " i = np.array(list(i.split())).astype(int)[::-1]\n", " # bring array in correct i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", " a = i[0:2]\n", @@ -73,29 +87,32 @@ " # compute y3 by y1+y2 mod 2 and save to array\n", "\n", " # compute overall winning condition -> alice at index of bob's row and vice versa\n", - " print(\"Alice's selection. Column:\"+str(col)+\"\\n\"+str(a.reshape(3,1))+\"\\n\")\n", - " print(\"Bob's circuit solution. Row:\"+str(row)+\"\\n\"+str(b)+\"\\n\")\n", + " #print(\"Alice's selection. Column:\"+str(col)+\"\\n\"+str(a.reshape(3,1))+\"\\n\")\n", + " #print(\"Bob's circuit solution. Row:\"+str(row)+\"\\n\"+str(b)+\"\\n\")\n", " \n", " if(winning_bob & winning_alice & (a[row-1]==b[col-1])):\n", - " print(\"Victory!\") \n", - " print(\"The following magic square wins the game!\")\n", + " #print(\"Victory!\") \n", + " #print(\"The following magic square wins the game!\")\n", + "\n", + " expected_states.append(str(i_orginal))\n", " k = np.empty((3,3,))\n", " k[:] = np.nan\n", " k[:,col-1] = a\n", " k[row-1,:] = b\n", - " print(k)\n", - " else:\n", - " print(\"The Game is lost!\")\n", - " print(\"------------------------------------------\") \n", - " print(\"\\n\")\n" + " #print(k)\n", + "\n", + " #print(\"------------------------------------------\") \n", + " #print(\"\\n\")\n", + " return expected_states" ] }, { "cell_type": "code", - "execution_count": 123, + "execution_count": 458, "metadata": {}, "outputs": [], "source": [ + "# function explaining the computation of x3, y3 and presenting the corresponding winning moves\n", "def explain(result, col, row):\n", " print(\"Resultant quantum circuit solution:\"+\"\\n\")\n", " print(result+\"\\n\")\n", @@ -131,12 +148,12 @@ }, { "cell_type": "code", - "execution_count": 124, + "execution_count": 368, "metadata": {}, "outputs": [], "source": [ - "\n", - "def get_state_accuracy(counts): # funciton to calculate state accuracy\n", + "# function to calculate state accuracy\n", + "def get_state_accuracy(counts, expected_states, shots): \n", " expected_counts = 0\n", " for state in counts.keys():\n", " if state in expected_states:\n", @@ -147,11 +164,11 @@ }, { "cell_type": "code", - "execution_count": 125, + "execution_count": 369, "metadata": {}, "outputs": [], "source": [ - "#The following function sets up the initial entanglement between Alice and Bob\n", + "# function that sets up the initial entanglement between Alice and Bob\n", "def share_bell_state(qc,a,b,c,d): \n", " qc.h(a)\n", " qc.h(b)\n", @@ -163,6 +180,7 @@ " if gamma==1:\n", " qc.h(a)\n", " qc.i(b)\n", + "\n", " elif gamma==2:\n", " qc.swap(a,b)\n", " qc.h(a)\n", @@ -177,8 +195,10 @@ " if gamma==1:\n", " qc.h(a)\n", " qc.h(b)\n", + "\n", " if gamma==2:\n", " qc.swap(a,b)\n", + "\n", " elif gamma==3:\n", " qc.z(a)\n", " qc.z(b)\n", @@ -187,9 +207,18 @@ " qc.h(b)" ] }, + { + "source": [ + "# Quantum Circuit of Mermim-Peres-Game\n", + "\n", + "#### Now the circuit modeling the Mirmim Peres game can be created and the user may select the respective column and row that ought to be played. Subsequently, the probability of winning with respect to the game moves of the circuit are shown and evaluated." + ], + "cell_type": "markdown", + "metadata": {} + }, { "cell_type": "code", - "execution_count": 127, + "execution_count": 455, "metadata": {}, "outputs": [ { @@ -199,207 +228,27 @@ "Alice and Bob, select a column or row value of 1, 2, or 3\n", "Alice choice (column):\n", "Bob choice (row):\n", - " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", - "q534_0: ┤ H ├──■────────░─┤ H ├─░─┤M├─────────\n", - " ├───┤ │ ░ ├───┤ ░ └╥┘┌─┐ \n", - "q534_1: ┤ H ├──┼────■───░─┤ I ├─░──╫─┤M├──────\n", - " └───┘┌─┴─┐ │ ░ ├───┤ ░ ║ └╥┘┌─┐ \n", - "q535_0: ─────┤ X ├──┼───░─┤ H ├─░──╫──╫─┤M├───\n", - " └───┘┌─┴─┐ ░ ├───┤ ░ ║ ║ └╥┘┌─┐\n", - "q535_1: ──────────┤ X ├─░─┤ H ├─░──╫──╫──╫─┤M├\n", - " └───┘ ░ └───┘ ░ ║ ║ ║ └╥┘\n", - "c160: 1/═══════════════════════════╩══╬══╬══╬═\n", - " 0 ║ ║ ║ \n", - " ║ ║ ║ \n", - "c161: 1/══════════════════════════════╩══╬══╬═\n", - " 0 ║ ║ \n", - " ║ ║ \n", - "c162: 1/═════════════════════════════════╩══╬═\n", - " 0 ║ \n", - " ║ \n", - "c163: 1/════════════════════════════════════╩═\n", - " 0 \n", - "Resultant quantum circuit solution:\n", - "\n", - "0 0 1 0\n", - "\n", - "Bring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", - "\n", - "[0 1 0 0]\n", - "\n", - "Compute missing x3 with x1+x2+1 mod 2 and save to array for further processing\n", - "Compute missing y3 with y1+y2 mod 2 and save to array for further processing\n", - "\n", - "Alice's column solution\n", - "[[0]\n", - " [1]\n", - " [0]]\n", - "Winning conditions fullfilled (Odd quantitude of 1)? :True\n", - "\n", - "Bob's row solution\n", - "[0 0 0]\n", - "Winning conditions fullfilled (Even quantitude of 1)? :True\n", - "\n", - "Create matrix to show results\n", - "\n", - "[[ 0. 0. 0.]\n", - " [ 1. nan nan]\n", - " [ 0. nan nan]]\n", - "------------------------------------------\n", - "------------------------------------------\n", - "Quantum circuit solution\n", - "0 0 1 0\n", - "\n", - "Alice's selection. Column:1\n", - "[[0]\n", - " [1]\n", - " [0]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[0 0 0]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "[[ 0. 0. 0.]\n", - " [ 1. nan nan]\n", - " [ 0. nan nan]]\n", - "------------------------------------------\n", - "\n", - "\n", - "Quantum circuit solution\n", - "0 1 1 1\n", - "\n", - "Alice's selection. Column:1\n", - "[[1]\n", - " [1]\n", - " [1]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[1 0 1]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "[[ 1. 0. 1.]\n", - " [ 1. nan nan]\n", - " [ 1. nan nan]]\n", - "------------------------------------------\n", - "\n", - "\n", - "Quantum circuit solution\n", - "0 1 0 1\n", - "\n", - "Alice's selection. Column:1\n", - "[[1]\n", - " [0]\n", - " [0]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[1 0 1]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "[[ 1. 0. 1.]\n", - " [ 0. nan nan]\n", - " [ 0. nan nan]]\n", - "------------------------------------------\n", - "\n", - "\n", - "Quantum circuit solution\n", - "1 0 1 0\n", - "\n", - "Alice's selection. Column:1\n", - "[[0]\n", - " [1]\n", - " [0]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[0 1 1]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "[[ 0. 1. 1.]\n", - " [ 1. nan nan]\n", - " [ 0. nan nan]]\n", - "------------------------------------------\n", - "\n", - "\n", - "Quantum circuit solution\n", - "1 1 0 1\n", - "\n", - "Alice's selection. Column:1\n", - "[[1]\n", - " [0]\n", - " [0]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[1 1 0]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "[[ 1. 1. 0.]\n", - " [ 0. nan nan]\n", - " [ 0. nan nan]]\n", - "------------------------------------------\n", - "\n", - "\n", - "Quantum circuit solution\n", - "0 0 0 0\n", - "\n", - "Alice's selection. Column:1\n", - "[[0]\n", - " [0]\n", - " [1]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[0 0 0]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "[[ 0. 0. 0.]\n", - " [ 0. nan nan]\n", - " [ 1. nan nan]]\n", - "------------------------------------------\n", - "\n", - "\n", - "Quantum circuit solution\n", - "1 1 1 1\n", - "\n", - "Alice's selection. Column:1\n", - "[[1]\n", - " [1]\n", - " [1]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[1 1 0]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "[[ 1. 1. 0.]\n", - " [ 1. nan nan]\n", - " [ 1. nan nan]]\n", - "------------------------------------------\n", - "\n", - "\n", - "Quantum circuit solution\n", - "1 0 0 0\n", - "\n", - "Alice's selection. Column:1\n", - "[[0]\n", - " [0]\n", - " [1]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[0 1 1]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "[[ 0. 1. 1.]\n", - " [ 0. nan nan]\n", - " [ 1. nan nan]]\n", - "------------------------------------------\n", - "\n", - "\n", - "{'0 0 1 0': 1041, '0 1 1 1': 1007, '0 1 0 1': 1033, '1 0 1 0': 1014, '1 1 0 1': 1039, '0 0 0 0': 1036, '1 1 1 1': 1028, '1 0 0 0': 994}\n" + " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", + "q2899_0: ┤ H ├──■────────░─┤ H ├─░─┤M├─────────\n", + " ├───┤ │ ░ ├───┤ ░ └╥┘┌─┐ \n", + "q2899_1: ┤ H ├──┼────■───░─┤ I ├─░──╫─┤M├──────\n", + " └───┘┌─┴─┐ │ ░ ├───┤ ░ ║ └╥┘┌─┐ \n", + "q2900_0: ─────┤ X ├──┼───░─┤ H ├─░──╫──╫─┤M├───\n", + " └───┘┌─┴─┐ ░ ├───┤ ░ ║ ║ └╥┘┌─┐\n", + "q2900_1: ──────────┤ X ├─░─┤ H ├─░──╫──╫──╫─┤M├\n", + " └───┘ ░ └───┘ ░ ║ ║ ║ └╥┘\n", + " c238: 1/═══════════════════════════╩══╬══╬══╬═\n", + " 0 ║ ║ ║ \n", + " ║ ║ ║ \n", + " c239: 1/══════════════════════════════╩══╬══╬═\n", + " 0 ║ ║ \n", + " ║ ║ \n", + " c240: 1/═════════════════════════════════╩══╬═\n", + " 0 ║ \n", + " ║ \n", + " c241: 1/════════════════════════════════════╩═\n", + " 0 \n", + "[('fake_yorktown', '51.9775390625%'), ('fake_melbourne', '91.015625%'), ('fake_vigo', '95.17822265625%'), ('fake_ourense', '94.32373046875%'), ('fake_valencia', '91.29638671875%'), ('fake_london', '91.357421875%'), ('fake_burlington', '91.6015625%'), ('fake_essex', '93.212890625%'), ('qasm_simulator', '100.0%'), ('qasm_simulator', '100.0%'), ('qasm_simulator', '100.0%')]\n" ] } ], @@ -438,24 +287,525 @@ " magicsquare_circuit.measure(3,3)\n", " \n", " magicsquare_circuit.draw(output='mpl')\n", - " print(magicsquare_circuit)\n", + " print(magicsquare_circuit) \n", + "\n", + " results_backends = []\n", + " backends.append(BasicAer.get_backend('qasm_simulator'))\n", + " for backend in backends:\n", + " job = execute(magicsquare_circuit, backend,optimization_level=3, shots=8192) # run the job simulation\n", + " results = job.result().get_counts()\n", + " expected_states = check_result(results, alpha, beta)\n", + " state_accuracy = get_state_accuracy(results,expected_states,8192)\n", + " results_backends.append((backend.name(), state_accuracy)) # store results\n", "\n", - " backend = BasicAer.get_backend('qasm_simulator') # define the backend\n", - " job = execute(magicsquare_circuit, backend, shots=8192) # run the job simulation\n", - " results = job.result().get_counts()\n", - " check_result(results, alpha, beta)\n", - " print(results)\n", + " print(results_backends)\n", "\n", "else:\n", " print(\"please enter values of 1, 2, or 3\")" ] }, + { + "source": [ + "#### Indeed, using an ideal quantum simulator, a winning probability of 1 can be achieved, exceeding the highest classical-winning probability of 8/9. Beyond that, however, it is desirable to quantify the results on a real IBM quantum computer, which will be realized in the following section. " + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [ + "# Circuit Performance on Real Quantum Devices\n", + "\n", + "#### Here we evaluate the performance of the ibmq_athens. Please note that you will have to add your own API token from the IBM Quantum website (https://quantum-computing.ibm.com)" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "cell_type": "code", + "execution_count": 53, + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "name": "stderr", + "text": [ + "configrc.store_credentials:WARNING:2021-03-15 09:28:25,642: Credentials already present. Set overwrite=True to overwrite.\n" + ] + } + ], + "source": [ + "IBMQ.save_account('YOUR_API_TOKEN') # retrieve from your IBM-Account: https://quantum-computing.ibm.com" + ] + }, + { + "cell_type": "code", + "execution_count": 392, + "metadata": {}, + "outputs": [], + "source": [ + "# load IBM Quantum Experience account \n", + "IBMQ.load_account()\n", + "provider = IBMQ.get_provider('ibm-q') # get provider" + ] + }, + { + "cell_type": "code", + "execution_count": 393, + "metadata": {}, + "outputs": [ + { + "output_type": "execute_result", + "data": { + "text/plain": [ + "[,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ]" + ] + }, + "metadata": {}, + "execution_count": 393 + } + ], + "source": [ + "provider.backends()" + ] + }, { "cell_type": "code", - "execution_count": null, + "execution_count": 394, "metadata": {}, "outputs": [], - "source": [] + "source": [ + "backend = provider.get_backend(\"ibmq_athens\")" + ] + }, + { + "cell_type": "code", + "execution_count": 395, + "metadata": {}, + "outputs": [], + "source": [ + "job = execute(magicsquare_circuit, backend,optimization_level=3, shots=8192) # run the job simulation with the highest compilter optimsation\n", + "results = job.result().get_counts()\n", + "expected_states = check_result(results, alpha, beta)\n", + "state_accuracy = get_state_accuracy(results,expected_states,8192)" + ] + }, + { + "cell_type": "code", + "execution_count": 396, + "metadata": {}, + "outputs": [ + { + "output_type": "execute_result", + "data": { + "text/plain": [ + "'96.69189453125%'" + ] + }, + "metadata": {}, + "execution_count": 396 + } + ], + "source": [ + "state_accuracy" + ] + }, + { + "source": [ + "#### Evidently, the MPG circuit outperforms the classical circuit by about 7.8% (88.89% vs. 96.69%) on a real, noisy, quantum computer." + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [ + "# Mathmatical background of the computation\n", + "\n", + "#### As we have demonstrated above, the quantum circuit solve the Mirmin-Peres game (MPG) on real quantum computers with a higher probability than with any classical strategy on ordinary computers. To shed some light on how and why this works, we present and explain the mathematical operations that lead to the solution of this game. In particular, we will first present a high-level equation describing the mathematical rationale, and we will then take a look at each element of this equation and finally explain it in one go using a concrete example.\n", + "\n", + "#### Recall that the quantum circuit has to play different moves for each different row/column instances of the MPG being selected, since there is no classical strategy that can win the game with probability = 1. Thus, to win with probability 1, the quantum circuit dynamically adapts to the input of each column and row played. Nevertheless, there is indeed one part of the circuit that remains static and is responsible for generating a matching qubit state and we will start of by focusing on that part. Abstractly, then, the quantum circuit can be described by the following formula: (For a technical background on why the gates were applied, see: [Quantum advantage in noisy flat circuits](https://www.nature.com/articles/s41567-020-0948-z)\n", + "\n", + "\n", + "$$ Quantum Solution MPG = Initial Vector * Static Matrix * Dynamic Matrix(row, column)$$\n", + "\n", + "\n", + "#### Starting with the first part of the equation, the static matrix & intial vector, of the MPG circuit is defined by quantum gates as follows:" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "cell_type": "code", + "execution_count": 403, + "metadata": {}, + "outputs": [ + { + "output_type": "execute_result", + "data": { + "text/plain": [ + "
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\n" + }, + "metadata": {}, + "execution_count": 403 + } + ], + "source": [ + "q = QuantumRegister(4) \n", + "c = ClassicalRegister(3) \n", + "qc = QuantumCircuit(q, c) \n", + "qc.h(q[0])\n", + "qc.h(q[1])\n", + "qc.cx(q[1],q[3])\n", + "qc.cx(q[0],q[2])\n", + "qc.barrier()\n", + "qc.draw(output='mpl')" + ] + }, + { + "source": [ + "### The initial vector state of the MPG circuit corresponds to the Dirac bracket |0000> and is hence described by the following vector: \n", + "$$ = {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}$$" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [ + "#### While the static matrix, represented as the quantum gates in the illustration above, is created by the appliance of two CNOT gates to the first and third qubit and the second and fourth qubit, as well as two Hadamard gates applied to the first and second qubit. Mathematically, this is described by the following equation: \n", + "\n", + "$$ Static Matrix= CNOT(2|4) ⨂ CNOT(1|3) * (I ⨂ I ⨂ H ⨂ H) $$\n", + "\n", + "Subsequently inserting the corresponding matrices into this equation results in the following matrix:" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [ + "$$ \n", + "StaticMatrix =\n", + "\n", + "\\begin{bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\ \n", + "\n", + "⨂\n", + "\n", + "\\begin {bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", + "\n", + "*\n", + "\n", + "\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} \n", + "\n", + "\n", + "$$" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [ + "$$ \n", + "StaticMatrix =\n", + "\n", + "\\begin{bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", + "*\n", + "\n", + "\\begin{bmatrix}\\ \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} \\\\\\end{bmatrix}\\\n", + "$$\n", + "\n", + "$$\n", + "=\\begin{bmatrix}\\ \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2}& 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} \\\\\\ \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", + "$$" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [ + "#### The matrix above is the final static matrix for each instance played with the MPG quantum circuit. This matrix can be further reduced by multplication with the, likewise static, inital state vector. Furthermore, this presents us with the Qubits considered for further usage in the dynmic parts of the circuit" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [ + "$$ = Static Matrix * Inital State $$\n", + "$$ \n", + "=\n", + "\\begin{bmatrix}\\ \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2}& 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} \\\\\\ \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", + "*\n", + "\\frac 1 {\\sqrt{2}} {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", + "$$\n", + "\n", + "$$ \n", + "= \n", + "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\ 0\\\\ \\frac 1 {2}\\end{pmatrix}\n", + "$$" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [ + "#### Then this vector holds the relevant qubits that can be further considered and modified / computed with in the dynmaic part of the circuit, these are:\n", + "$$= \\frac 1 {2} (|0000\\rangle+|0101\\rangle+|1010\\rangle+|1111\\rangle)$$\n", + "#### Moreover, due to the nature of matrix multiplication, this vector allows the following simplification. For each row of the first matrix, we only need to consider the numbers marked in red and calculate their product sum with the static vector on the right, as the other positions are multiplied by zero." + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [ + "$$ = \n", + "\n", + "\\begin{bmatrix}\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\end{bmatrix}\\ \n", + "\n", + "*\n", + "\n", + "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\ 0\\\\ \\frac 1 {2}\\end{pmatrix}\n", + "$$" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [ + "### Now, in order to consider the later, dynamic, part of the equation we will have to select an example. In this case we have selected row: 1 and column: 1. The corresponding quantum circuit is as follows:" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "cell_type": "code", + "execution_count": 439, + "metadata": {}, + "outputs": [ + { + "output_type": "execute_result", + "data": { + "text/plain": [ + "
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\n" + }, + "metadata": {}, + "execution_count": 439 + } + ], + "source": [ + "q = QuantumRegister(4) \n", + "c = ClassicalRegister(3) \n", + "qc = QuantumCircuit(q, c) \n", + "#qc.h(q[0])\n", + "#qc.h(q[1])\n", + "#qc.cx(q[1],q[3])\n", + "#qc.cx(q[0],q[2])\n", + "qc.barrier()\n", + "qc.h(q[0])\n", + "qc.id(q[1])\n", + "qc.h(q[2])\n", + "qc.h(q[3])\n", + "qc.draw(output='mpl')" + ] + }, + { + "source": [ + "#### The translation of these circuit gates into mathematical matrices is in turn equivalent to the following sequence of operations:\n", + "$$ = H ⨂ H ⨂ I ⨂ H $$\n", + "$$\n", + "= \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix}\n", + "$$\n", + "$$\n", + "DynamicMatrix(1, 1) =\n", + "\\begin{bmatrix}\\ \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", + "\\\\\\\n", + " 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", + "\\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", + "\\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", + "\\\\\\\n", + " \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0\n", + " \\\\\\\n", + " \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0\n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}\n", + " \\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0\n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \\\\\\end{bmatrix}\\ \n", + "$$" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [ + "#### The matrix above is then the one employed when row / column: 1 is selected. In order to now ascertain the winning combinations for that particular game, we have to multiply the dynamic matrix with the static." + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [ + " $$\n", + "Winning Plays=\n", + "\\begin{bmatrix}\\ \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", + "\\\\\\\n", + " 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", + "\\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", + "\\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", + "\\\\\\\n", + " \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0\n", + " \\\\\\\n", + " \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0\n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}\n", + " \\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0\n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \\\\\\end{bmatrix}\\ \n", + "\n", + "*\n", + "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\ 0\\\\ \\frac 1 {2}\\end{pmatrix}\n", + "$$\n", + "\n", + "$$\n", + "= \\begin{pmatrix} \\frac 1 {\\sqrt{8}} \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\ -\\frac 1 {\\sqrt{8}} \\\\0 \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\ 0\\\\ -\\frac 1 {\\sqrt{8}} \\end{pmatrix}\n", + "$$\n" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [ + "#### Finally, this vector contains the qubits that can be played to win the game with probability 1, these are:\n", + "\n", + "$$= \\frac 1 {8} (|0000\\rangle+|0100\\rangle+|0101\\rangle+|01111\\rangle+|1000\\rangle)+|1011\\rangle)+|1101\\rangle)+|1111\\rangle)$$\n", + "\n", + "#### To validate these mathematical results, we will run the quantum circuit in one go and compare the respective qubit results" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "cell_type": "code", + "execution_count": 460, + "metadata": {}, + "outputs": [ + { + "output_type": "execute_result", + "data": { + "text/plain": [ + "
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\n" + }, + "metadata": {}, + "execution_count": 460 + } + ], + "source": [ + "q = QuantumRegister(4) \n", + "c = ClassicalRegister(4) \n", + "qc = QuantumCircuit(q, c) \n", + "qc.h(q[0])\n", + "qc.h(q[1])\n", + "qc.cx(q[1],q[3])\n", + "qc.cx(q[0],q[2])\n", + "qc.barrier()\n", + "qc.h(q[0])\n", + "qc.id(q[1])\n", + "qc.h(q[2])\n", + "qc.h(q[3])\n", + "qc.measure(0,0)\n", + "qc.measure(1,1)\n", + "qc.measure(2,2)\n", + "qc.measure(3,3)\n", + "qc.draw(output='mpl')" + ] + }, + { + "cell_type": "code", + "execution_count": 457, + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "{'0000': 1027, '0010': 1062, '0101': 1032, '0111': 993, '1000': 1047, '1010': 1046, '1101': 975, '1111': 1010}\nAccuracy: 100.0%\n" + ] + } + ], + "source": [ + "backend = Aer.get_backend('qasm_simulator')\n", + "result = execute(qc, backend,shots=8192).result().get_counts()\n", + "expected_states = check_result(results, 1, 1)\n", + "state_accuracy = get_state_accuracy(results,expected_states,8192)\n", + "print(result)\n", + "print(\"Accuracy: \"+state_accuracy)" + ] + }, + { + "source": [ + "#### Evidently, the results of the mathematical calculation match those of the circuit and an win rate of 100% is achieved!" + ], + "cell_type": "markdown", + "metadata": {} } ] } \ No newline at end of file From c064ec33fb0df4f3108ae6be308f71213266eb1b Mon Sep 17 00:00:00 2001 From: David-Drexlin <39337061+David-Drexlin@users.noreply.github.com> Date: Tue, 23 Mar 2021 22:58:23 +0100 Subject: [PATCH 10/30] Delete .vscode directory --- .vscode/settings.json | 3 --- 1 file changed, 3 deletions(-) delete mode 100644 .vscode/settings.json diff --git a/.vscode/settings.json b/.vscode/settings.json deleted file mode 100644 index 625b81c..0000000 --- a/.vscode/settings.json +++ /dev/null @@ -1,3 +0,0 @@ -{ - "jupyter.jupyterServerType": "local" -} \ No newline at end of file From 17355e11b25dde1a55bd64341fabfb247ab35060 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Thu, 25 Mar 2021 10:43:43 +0100 Subject: [PATCH 11/30] fix: Names, Imports, Chronology --- "Mermin\342\200\223Peres-Game.ipynb" | 2266 +++++++++++++++++++++++--- 1 file changed, 2006 insertions(+), 260 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 94bde9d..8e45e87 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -1,59 +1,36 @@ { - "metadata": { - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.9.1-final" - }, - "orig_nbformat": 2, - "kernelspec": { - "name": "python3", - "display_name": "Python 3", - "language": "python" - } - }, - "nbformat": 4, - "nbformat_minor": 2, "cells": [ { + "cell_type": "markdown", + "metadata": {}, "source": [ "# Mermin Peres Magic Square Game" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "### In order to solve and test the Mirmin Peres Magic Square (MPG) game on a quantum computer, we will employ the Qiskit framework and the IBM Q-Experience. Subsequently, this notebook will introduce the quantum circuit solving the MPG game, quantify it's performance on both simulated and real devices, and finally provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game.\n", "\n", "#### For the subsequent circuit definition, evaluation and illustration, we first have to define auxiliary functions and import the corresponding modules from Qiskit and Numpy" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { "cell_type": "code", - "execution_count": 365, + "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "from qiskit import *\n", "import numpy as np\n", "from qiskit.test.mock import *\n", - "from qiskit.visualization import plot_state_qsphere\n", - "from qiskit_textbook.tools import array_to_latex" + "from qiskit.visualization import plot_state_qsphere" ] }, { "cell_type": "code", - "execution_count": 366, + "execution_count": 2, "metadata": {}, "outputs": [], "source": [ @@ -108,7 +85,7 @@ }, { "cell_type": "code", - "execution_count": 458, + "execution_count": 3, "metadata": {}, "outputs": [], "source": [ @@ -148,7 +125,7 @@ }, { "cell_type": "code", - "execution_count": 368, + "execution_count": 4, "metadata": {}, "outputs": [], "source": [ @@ -164,7 +141,7 @@ }, { "cell_type": "code", - "execution_count": 369, + "execution_count": 5, "metadata": {}, "outputs": [], "source": [ @@ -207,48 +184,39 @@ " qc.h(b)" ] }, - { - "source": [ - "# Quantum Circuit of Mermim-Peres-Game\n", - "\n", - "#### Now the circuit modeling the Mirmim Peres game can be created and the user may select the respective column and row that ought to be played. Subsequently, the probability of winning with respect to the game moves of the circuit are shown and evaluated." - ], - "cell_type": "markdown", - "metadata": {} - }, { "cell_type": "code", - "execution_count": 455, + "execution_count": 14, "metadata": {}, "outputs": [ { - "output_type": "stream", "name": "stdout", + "output_type": "stream", "text": [ "Alice and Bob, select a column or row value of 1, 2, or 3\n", "Alice choice (column):\n", "Bob choice (row):\n", - " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", - "q2899_0: ┤ H ├──■────────░─┤ H ├─░─┤M├─────────\n", - " ├───┤ │ ░ ├───┤ ░ └╥┘┌─┐ \n", - "q2899_1: ┤ H ├──┼────■───░─┤ I ├─░──╫─┤M├──────\n", - " └───┘┌─┴─┐ │ ░ ├───┤ ░ ║ └╥┘┌─┐ \n", - "q2900_0: ─────┤ X ├──┼───░─┤ H ├─░──╫──╫─┤M├───\n", - " └───┘┌─┴─┐ ░ ├───┤ ░ ║ ║ └╥┘┌─┐\n", - "q2900_1: ──────────┤ X ├─░─┤ H ├─░──╫──╫──╫─┤M├\n", - " └───┘ ░ └───┘ ░ ║ ║ ║ └╥┘\n", - " c238: 1/═══════════════════════════╩══╬══╬══╬═\n", - " 0 ║ ║ ║ \n", - " ║ ║ ║ \n", - " c239: 1/══════════════════════════════╩══╬══╬═\n", - " 0 ║ ║ \n", - " ║ ║ \n", - " c240: 1/═════════════════════════════════╩══╬═\n", - " 0 ║ \n", - " ║ \n", - " c241: 1/════════════════════════════════════╩═\n", - " 0 \n", - "[('fake_yorktown', '51.9775390625%'), ('fake_melbourne', '91.015625%'), ('fake_vigo', '95.17822265625%'), ('fake_ourense', '94.32373046875%'), ('fake_valencia', '91.29638671875%'), ('fake_london', '91.357421875%'), ('fake_burlington', '91.6015625%'), ('fake_essex', '93.212890625%'), ('qasm_simulator', '100.0%'), ('qasm_simulator', '100.0%'), ('qasm_simulator', '100.0%')]\n" + " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", + "q163_0: ┤ H ├──■────────░─┤ H ├─░─┤M├─────────\n", + " ├───┤ │ ░ ├───┤ ░ └╥┘┌─┐ \n", + "q163_1: ┤ H ├──┼────■───░─┤ I ├─░──╫─┤M├──────\n", + " └───┘┌─┴─┐ │ ░ ├───┤ ░ ║ └╥┘┌─┐ \n", + "q164_0: ─────┤ X ├──┼───░─┤ H ├─░──╫──╫─┤M├───\n", + " └───┘┌─┴─┐ ░ ├───┤ ░ ║ ║ └╥┘┌─┐\n", + "q164_1: ──────────┤ X ├─░─┤ H ├─░──╫──╫──╫─┤M├\n", + " └───┘ ░ └───┘ ░ ║ ║ ║ └╥┘\n", + " c20: 1/═══════════════════════════╩══╬══╬══╬═\n", + " 0 ║ ║ ║ \n", + " ║ ║ ║ \n", + " c21: 1/══════════════════════════════╩══╬══╬═\n", + " 0 ║ ║ \n", + " ║ ║ \n", + " c22: 1/═════════════════════════════════╩══╬═\n", + " 0 ║ \n", + " ║ \n", + " c23: 1/════════════════════════════════════╩═\n", + " 0 \n", + "[('fake_yorktown', '51.86767578125%'), ('fake_melbourne', '91.73583984375%'), ('fake_vigo', '95.41015625%'), ('fake_ourense', '94.27490234375%'), ('fake_valencia', '91.14990234375%'), ('fake_london', '92.05322265625%'), ('fake_burlington', '91.5771484375%'), ('fake_essex', '93.603515625%'), ('qasm_simulator', '100.0%')]\n" ] } ], @@ -290,7 +258,10 @@ " print(magicsquare_circuit) \n", "\n", " results_backends = []\n", - " backends.append(BasicAer.get_backend('qasm_simulator'))\n", + "\n", + " backends = [FakeYorktown(), FakeMelbourne(), FakeVigo(), FakeOurense(), FakeValencia(), FakeLondon(),\n", + " FakeBurlington(), FakeEssex(), BasicAer.get_backend('qasm_simulator')]\n", + "\n", " for backend in backends:\n", " job = execute(magicsquare_circuit, backend,optimization_level=3, shots=8192) # run the job simulation\n", " results = job.result().get_counts()\n", @@ -305,130 +276,24 @@ ] }, { - "source": [ - "#### Indeed, using an ideal quantum simulator, a winning probability of 1 can be achieved, exceeding the highest classical-winning probability of 8/9. Beyond that, however, it is desirable to quantify the results on a real IBM quantum computer, which will be realized in the following section. " - ], - "cell_type": "markdown", - "metadata": {} - }, - { - "source": [ - "# Circuit Performance on Real Quantum Devices\n", - "\n", - "#### Here we evaluate the performance of the ibmq_athens. Please note that you will have to add your own API token from the IBM Quantum website (https://quantum-computing.ibm.com)" - ], "cell_type": "markdown", - "metadata": {} - }, - { - "cell_type": "code", - "execution_count": 53, - "metadata": {}, - "outputs": [ - { - "output_type": "stream", - "name": "stderr", - "text": [ - "configrc.store_credentials:WARNING:2021-03-15 09:28:25,642: Credentials already present. Set overwrite=True to overwrite.\n" - ] - } - ], - "source": [ - "IBMQ.save_account('YOUR_API_TOKEN') # retrieve from your IBM-Account: https://quantum-computing.ibm.com" - ] - }, - { - "cell_type": "code", - "execution_count": 392, - "metadata": {}, - "outputs": [], - "source": [ - "# load IBM Quantum Experience account \n", - "IBMQ.load_account()\n", - "provider = IBMQ.get_provider('ibm-q') # get provider" - ] - }, - { - "cell_type": "code", - "execution_count": 393, - "metadata": {}, - "outputs": [ - { - "output_type": "execute_result", - "data": { - "text/plain": [ - "[,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ]" - ] - }, - "metadata": {}, - "execution_count": 393 - } - ], - "source": [ - "provider.backends()" - ] - }, - { - "cell_type": "code", - "execution_count": 394, "metadata": {}, - "outputs": [], "source": [ - "backend = provider.get_backend(\"ibmq_athens\")" - ] - }, - { - "cell_type": "code", - "execution_count": 395, - "metadata": {}, - "outputs": [], - "source": [ - "job = execute(magicsquare_circuit, backend,optimization_level=3, shots=8192) # run the job simulation with the highest compilter optimsation\n", - "results = job.result().get_counts()\n", - "expected_states = check_result(results, alpha, beta)\n", - "state_accuracy = get_state_accuracy(results,expected_states,8192)" + "# Quantum Circuit of Mermim-Peres-Game\n", + "\n", + "#### Now the circuit modeling the Mirmim Peres game can be created and the user may select the respective column and row that ought to be played. Subsequently, the probability of winning with respect to the game moves of the circuit are shown and evaluated." ] }, { - "cell_type": "code", - "execution_count": 396, + "cell_type": "markdown", "metadata": {}, - "outputs": [ - { - "output_type": "execute_result", - "data": { - "text/plain": [ - "'96.69189453125%'" - ] - }, - "metadata": {}, - "execution_count": 396 - } - ], "source": [ - "state_accuracy" + "#### Indeed, using an ideal quantum simulator, a winning probability of 1 can be achieved, exceeding the highest classical-winning probability of 8/9. Beyond that, however, it is desirable to quantify the results on a real IBM quantum computer, which will be realized in the last section. " ] }, { - "source": [ - "#### Evidently, the MPG circuit outperforms the classical circuit by about 7.8% (88.89% vs. 96.69%) on a real, noisy, quantum computer." - ], "cell_type": "markdown", - "metadata": {} - }, - { + "metadata": {}, "source": [ "# Mathmatical background of the computation\n", "\n", @@ -437,13 +302,11 @@ "#### Recall that the quantum circuit has to play different moves for each different row/column instances of the MPG being selected, since there is no classical strategy that can win the game with probability = 1. Thus, to win with probability 1, the quantum circuit dynamically adapts to the input of each column and row played. Nevertheless, there is indeed one part of the circuit that remains static and is responsible for generating a matching qubit state and we will start of by focusing on that part. Abstractly, then, the quantum circuit can be described by the following formula: (For a technical background on why the gates were applied, see: [Quantum advantage in noisy flat circuits](https://www.nature.com/articles/s41567-020-0948-z)\n", "\n", "\n", - "$$ Quantum Solution MPG = Initial Vector * Static Matrix * Dynamic Matrix(row, column)$$\n", + "$$ Quantum Solution MPG = Initial Vector * Dynamic Matrix(row, column)$$\n", "\n", "\n", - "#### Starting with the first part of the equation, the static matrix & intial vector, of the MPG circuit is defined by quantum gates as follows:" - ], - "cell_type": "markdown", - "metadata": {} + "#### Starting with the first part of the equation, the intial vector, of the MPG circuit is defined by quantum gates as follows:" + ] }, { "cell_type": "code", @@ -451,16 +314,549 @@ "metadata": {}, "outputs": [ { - "output_type": "execute_result", "data": { + "image/png": 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\n" + ] }, + "execution_count": 403, "metadata": {}, - "execution_count": 403 + "output_type": "execute_result" } ], "source": [ @@ -476,30 +872,21 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ - "### The initial vector state of the MPG circuit corresponds to the Dirac bracket |0000> and is hence described by the following vector: \n", + "#### The very starting vector state of the MPG circuit, prior to any quantum gate interaction, corresponds to the Dirac bracket |0000> and is hence described by the following vector: \n", "$$ = {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}$$" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { - "source": [], "cell_type": "markdown", - "metadata": {} + "metadata": {}, + "source": [] }, { - "source": [ - "#### While the static matrix, represented as the quantum gates in the illustration above, is created by the appliance of two CNOT gates to the first and third qubit and the second and fourth qubit, as well as two Hadamard gates applied to the first and second qubit. Mathematically, this is described by the following equation: \n", - "\n", - "$$ Static Matrix= CNOT(2|4) ⨂ CNOT(1|3) * (I ⨂ I ⨂ H ⨂ H) $$\n", - "\n", - "Subsequently inserting the corresponding matrices into this equation results in the following matrix:" - ], "cell_type": "markdown", - "metadata": {} - }, - { + "metadata": {}, "source": [ "$$ \n", "StaticMatrix =\n", @@ -516,16 +903,27 @@ "\n", "\n", "$$" - ], + ] + }, + { "cell_type": "markdown", - "metadata": {} + "metadata": {}, + "source": [ + "#### While the static matrix, represented as the quantum gates in the illustration above, is created by the appliance of two CNOT gates to the first and third qubit and the second and fourth qubit, as well as two Hadamard gates applied to the first and second qubit. Mathematically, this is described by the following equation: \n", + "\n", + "$$ Static Matrix= CNOT_{2|4} ⨂ CNOT_{1|3} * (I ⨂ I ⨂ H ⨂ H) $$\n", + "\n", + "Subsequently inserting the corresponding matrices into this equation results in the following matrix:" + ] }, { - "source": [], "cell_type": "markdown", - "metadata": {} + "metadata": {}, + "source": [] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "$$ \n", "StaticMatrix =\n", @@ -539,18 +937,18 @@ "$$\n", "=\\begin{bmatrix}\\ \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2}& 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} \\\\\\ \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", "$$" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "#### The matrix above is the final static matrix for each instance played with the MPG quantum circuit. This matrix can be further reduced by multplication with the, likewise static, inital state vector. Furthermore, this presents us with the Qubits considered for further usage in the dynmic parts of the circuit" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "$$ = Static Matrix * Inital State $$\n", "$$ \n", @@ -564,20 +962,20 @@ "= \n", "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\ 0\\\\ \\frac 1 {2}\\end{pmatrix}\n", "$$" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "#### Then this vector holds the relevant qubits that can be further considered and modified / computed with in the dynmaic part of the circuit, these are:\n", "$$= \\frac 1 {2} (|0000\\rangle+|0101\\rangle+|1010\\rangle+|1111\\rangle)$$\n", "#### Moreover, due to the nature of matrix multiplication, this vector allows the following simplification. For each row of the first matrix, we only need to consider the numbers marked in red and calculate their product sum with the static vector on the right, as the other positions are multiplied by zero." - ], - "cell_type": "markdown", - "metadata": {} + ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "$$ = \n", "\n", @@ -587,16 +985,7 @@ "\n", "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\ 0\\\\ \\frac 1 {2}\\end{pmatrix}\n", "$$" - ], - "cell_type": "markdown", - "metadata": {} - }, - { - "source": [ - "### Now, in order to consider the later, dynamic, part of the equation we will have to select an example. In this case we have selected row: 1 and column: 1. The corresponding quantum circuit is as follows:" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { "cell_type": "code", @@ -604,16 +993,471 @@ "metadata": {}, "outputs": [ { - "output_type": "execute_result", "data": { + "image/png": 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\n" + ] }, + "execution_count": 439, "metadata": {}, - "execution_count": 439 + "output_type": "execute_result" } ], "source": [ @@ -633,6 +1477,15 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Now, in order to consider the later, dynamic, part of the equation we will have to select an example. In this case we have selected row: 1 and column: 1. The corresponding quantum circuit is as follows:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, "source": [ "#### The translation of these circuit gates into mathematical matrices is in turn equivalent to the following sequence of operations:\n", "$$ = H ⨂ H ⨂ I ⨂ H $$\n", @@ -672,18 +1525,18 @@ " \\\\\\ \n", "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \\\\\\end{bmatrix}\\ \n", "$$" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "#### The matrix above is then the one employed when row / column: 1 is selected. In order to now ascertain the winning combinations for that particular game, we have to multiply the dynamic matrix with the static." - ], - "cell_type": "markdown", - "metadata": {} + ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ " $$\n", "Winning Plays=\n", @@ -725,20 +1578,18 @@ "$$\n", "= \\begin{pmatrix} \\frac 1 {\\sqrt{8}} \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\ -\\frac 1 {\\sqrt{8}} \\\\0 \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\ 0\\\\ -\\frac 1 {\\sqrt{8}} \\end{pmatrix}\n", "$$\n" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "#### Finally, this vector contains the qubits that can be played to win the game with probability 1, these are:\n", "\n", "$$= \\frac 1 {8} (|0000\\rangle+|0100\\rangle+|0101\\rangle+|01111\\rangle+|1000\\rangle)+|1011\\rangle)+|1101\\rangle)+|1111\\rangle)$$\n", "\n", "#### To validate these mathematical results, we will run the quantum circuit in one go and compare the respective qubit results" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { "cell_type": "code", @@ -746,16 +1597,772 @@ "metadata": {}, "outputs": [ { - "output_type": "execute_result", "data": { + "image/png": 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\n" + ] }, + "execution_count": 460, "metadata": {}, - "execution_count": 460 + "output_type": "execute_result" } ], "source": [ @@ -784,10 +2391,11 @@ "metadata": {}, "outputs": [ { - "output_type": "stream", "name": "stdout", + "output_type": "stream", "text": [ - "{'0000': 1027, '0010': 1062, '0101': 1032, '0111': 993, '1000': 1047, '1010': 1046, '1101': 975, '1111': 1010}\nAccuracy: 100.0%\n" + "{'0000': 1027, '0010': 1062, '0101': 1032, '0111': 993, '1000': 1047, '1010': 1046, '1101': 975, '1111': 1010}\n", + "Accuracy: 100.0%\n" ] } ], @@ -801,11 +2409,149 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Evidently, the results of the mathematical calculation match those of the circuit and a win rate of 100% is achieved!" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Circuit Performance on Real Quantum Devices\n", + "\n", + "#### Here we evaluate the performance of the ibmq_athens. Please note that you will have to add your own API token from the IBM Quantum website (https://quantum-computing.ibm.com)" + ] + }, + { + "cell_type": "code", + "execution_count": 53, + "metadata": {}, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "configrc.store_credentials:WARNING:2021-03-15 09:28:25,642: Credentials already present. Set overwrite=True to overwrite.\n" + ] + } + ], + "source": [ + "IBMQ.save_account('YOUR_API_TOKEN') # retrieve from your IBM-Account: https://quantum-computing.ibm.com" + ] + }, + { + "cell_type": "code", + "execution_count": 392, + "metadata": {}, + "outputs": [], + "source": [ + "# load IBM Quantum Experience account \n", + "IBMQ.load_account()\n", + "provider = IBMQ.get_provider('ibm-q') # get provider" + ] + }, + { + "cell_type": "code", + "execution_count": 393, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ,\n", + " ]" + ] + }, + "execution_count": 393, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "provider.backends()" + ] + }, + { + "cell_type": "code", + "execution_count": 394, + "metadata": {}, + "outputs": [], + "source": [ + "backend = provider.get_backend(\"ibmq_athens\")" + ] + }, + { + "cell_type": "code", + "execution_count": 395, + "metadata": {}, + "outputs": [], "source": [ - "#### Evidently, the results of the mathematical calculation match those of the circuit and an win rate of 100% is achieved!" + "job = execute(magicsquare_circuit, backend,optimization_level=3, shots=8192) # run the job simulation with the highest compilter optimsation\n", + "results = job.result().get_counts()\n", + "expected_states = check_result(results, alpha, beta)\n", + "state_accuracy = get_state_accuracy(results,expected_states,8192)" + ] + }, + { + "cell_type": "code", + "execution_count": 396, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "'96.69189453125%'" + ] + }, + "execution_count": 396, + "metadata": {}, + "output_type": "execute_result" + } ], + "source": [ + "state_accuracy" + ] + }, + { "cell_type": "markdown", - "metadata": {} + "metadata": {}, + "source": [ + "#### Evidently, the MPG circuit outperforms the classical circuit by about 7.8% (88.89% vs. 96.69%) on a real, noisy, quantum computer." + ] } - ] -} \ No newline at end of file + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.5" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} From 7b48dac3280a528ad503a3680526c353da0cf32b Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Thu, 25 Mar 2021 10:44:14 +0100 Subject: [PATCH 12/30] fix: Names, Imports, Chronology --- "Mermin\342\200\223Peres-Game.ipynb" | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 8e45e87..04d3d8d 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -11,7 +11,8 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "### In order to solve and test the Mirmin Peres Magic Square (MPG) game on a quantum computer, we will employ the Qiskit framework and the IBM Q-Experience. Subsequently, this notebook will introduce the quantum circuit solving the MPG game, quantify it's performance on both simulated and real devices, and finally provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game.\n", + "### In order to solve and test the Mirmin Peres Magic Square (MPG) game on a quantum computer, we \n", + "employ the Qiskit framework and the IBM Q-Experience. Subsequently, this notebook will introduce the quantum circuit solving the MPG game, quantify it's performance on both simulated and real devices, and finally provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game.\n", "\n", "#### For the subsequent circuit definition, evaluation and illustration, we first have to define auxiliary functions and import the corresponding modules from Qiskit and Numpy" ] @@ -2554,4 +2555,4 @@ }, "nbformat": 4, "nbformat_minor": 2 -} +} \ No newline at end of file From 29f83e0f95b1edc3f7930bb46b56297e3d251a64 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Thu, 25 Mar 2021 11:21:15 +0100 Subject: [PATCH 13/30] fix: list to array --- "Mermin\342\200\223Peres-Game.ipynb" | 50 +++++++++------------------- 1 file changed, 16 insertions(+), 34 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index ffbf598..053c553 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -1,26 +1,4 @@ { - "metadata": { - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5-final" - }, - "orig_nbformat": 2, - "kernelspec": { - "name": "python3", - "display_name": "Python 3", - "language": "python" - } - }, - "nbformat": 4, - "nbformat_minor": 2, "cells": [ { "cell_type": "markdown", @@ -59,15 +37,19 @@ "source": [ "#checks if the circuit output fulfills the winning criteria\n", "def check_result(results, col, row):\n", + " \n", " print_tut = True\n", + " expected_states = []\n", + " \n", " for i in results: \n", " \n", - " if print_tut:\n", - " explain(i, col, row)\n", - " print_tut = False\n", + " #if print_tut:\n", + " # explain(i, col, row)\n", + " # print_tut = False\n", "\n", - " print(\"Quantum circuit solution\")\n", - " print(i+\"\\n\")\n", + " #print(\"Quantum circuit solution\")\n", + " #print(i+\"\\n\")\n", + " i_orginal = i\n", " i = np.array(list(i.split())).astype(int)[::-1]\n", " # bring array in correct i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", " a = i[0:2]\n", @@ -85,8 +67,8 @@ " # compute y3 by y1+y2 mod 2 and save to array\n", "\n", " # compute overall winning condition -> alice at index of bob's row and vice versa\n", - " print(\"Alice's selection. Column:\"+str(col)+\"\\n\"+str(a.reshape(3,1))+\"\\n\")\n", - " print(\"Bob's circuit solution. Row:\"+str(row)+\"\\n\"+str(b)+\"\\n\")\n", + " #print(\"Alice's selection. Column:\"+str(col)+\"\\n\"+str(a.reshape(3,1))+\"\\n\")\n", + " #print(\"Bob's circuit solution. Row:\"+str(row)+\"\\n\"+str(b)+\"\\n\")\n", " \n", " if(winning_bob & winning_alice & (a[row-1]==b[col-1])):\n", " #print(\"Victory!\") \n", @@ -97,11 +79,11 @@ " k[:] = np.nan\n", " k[:,col-1] = a\n", " k[row-1,:] = b\n", - " print(k)\n", - " else:\n", - " print(\"The Game is lost!\")\n", - " print(\"------------------------------------------\") \n", - " print(\"\\n\")\n" + " #print(k)\n", + "\n", + " #print(\"------------------------------------------\") \n", + " #print(\"\\n\")\n", + " return expected_states" ] }, { From 6f8384a58ee8fe51afc4373dc35d7b1f3c83844b Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Thu, 25 Mar 2021 13:37:13 +0100 Subject: [PATCH 14/30] fix: display of math tex --- "Mermin\342\200\223Peres-Game.ipynb" | 15 ++++----------- 1 file changed, 4 insertions(+), 11 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 053c553..b96e942 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -955,8 +955,7 @@ "source": [ "$$ = Static Matrix * Inital State $$\n", "$$ \n", - "=\n", - "\\begin{bmatrix}\\ \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2}& 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} \\\\\\ \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", + "= \\begin{bmatrix}\\ \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2}& 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} \\\\\\ \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", "*\n", "\\frac 1 {\\sqrt{2}} {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", "$$\n", @@ -981,11 +980,8 @@ "metadata": {}, "source": [ "$$ = \n", - "\n", - "\\begin{bmatrix}\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\end{bmatrix}\\ \n", - "\n", + "\\begin{bmatrix}\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\end{bmatrix} \n", "*\n", - "\n", "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\ 0\\\\ \\frac 1 {2}\\end{pmatrix}\n", "$$" ] @@ -1542,8 +1538,7 @@ "metadata": {}, "source": [ " $$\n", - "Winning Plays=\n", - "\\begin{bmatrix}\\ \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \\\\\\ \n", + "Winning Plays= \\begin{bmatrix}\\ \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \\\\\\ \n", "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 \n", "\\\\\\ \n", "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", @@ -1573,11 +1568,9 @@ "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", " \\\\\\ \n", "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \\\\\\end{bmatrix}\\ \n", - "\n", "*\n", "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\ 0\\\\ \\frac 1 {2}\\end{pmatrix}\n", "$$\n", - "\n", "$$\n", "= \\begin{pmatrix} \\frac 1 {\\sqrt{8}} \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\ -\\frac 1 {\\sqrt{8}} \\\\0 \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\ 0\\\\ -\\frac 1 {\\sqrt{8}} \\end{pmatrix}\n", "$$\n" @@ -2557,4 +2550,4 @@ }, "nbformat": 4, "nbformat_minor": 2 -} \ No newline at end of file +} From c5165fc30483bb000c6ebecf97ca19f97df88dc8 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Thu, 25 Mar 2021 13:57:36 +0100 Subject: [PATCH 15/30] merge --- "Mermin\342\200\223Peres-Game.ipynb" | 27 ++------------------------- 1 file changed, 2 insertions(+), 25 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index b96e942..bc6e878 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -882,29 +882,14 @@ "$$ = {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}$$" ] }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [] - }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$ \n", - "StaticMatrix =\n", - "\n", - "\\begin{bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\ \n", - "\n", - "⨂\n", - "\n", - "\\begin {bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", - "\n", + "StaticMatrix = \\begin{bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\ ⨂ \\begin {bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", "*\n", - "\n", "\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} \n", - "\n", - "\n", "$$" ] }, @@ -919,21 +904,13 @@ "Subsequently inserting the corresponding matrices into this equation results in the following matrix:" ] }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [] - }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$ \n", - "StaticMatrix =\n", - "\n", - "\\begin{bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", + "StaticMatrix = \\begin{bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", "*\n", - "\n", "\\begin{bmatrix}\\ \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} \\\\\\end{bmatrix}\\\n", "$$\n", "\n", From dfd6f32904787174d3a22307b36967fe7c347620 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Tue, 30 Mar 2021 10:59:39 +0200 Subject: [PATCH 16/30] fix: Jan's Change Request --- "Mermin\342\200\223Peres-Game.ipynb" | 2379 ++++++---------------- Mirmin-Peres-Game/Circuit_Operations.png | Bin 0 -> 77515 bytes Mirmin-Peres-Game/MPG.png | Bin 0 -> 4833 bytes Mirmin-Peres-Game/Pauli_Operators.png | Bin 0 -> 26984 bytes 4 files changed, 601 insertions(+), 1778 deletions(-) create mode 100644 Mirmin-Peres-Game/Circuit_Operations.png create mode 100644 Mirmin-Peres-Game/MPG.png create mode 100644 Mirmin-Peres-Game/Pauli_Operators.png diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index bc6e878..b42b8f9 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -8,30 +8,44 @@ ] }, { + "source": [ + "The Mermin Peres Magic Square Game is a Game that is not solvable with a probability of 100% on classical computers. It operates in the following manner: Two players, Alice and Bob, play a cooperative game, yet unable to communicat by playing in a respectivly randomly assigned column / row either +1 or -1. The game is won if the number of +1 played in the selcted column is even and the -1 played in the rows is odd, further and crucially, the intersecting cell between column and row must be the same. As illustrated in the image below there is no classical solution to win the game with probability of 100%, however, the game can be won with a probability of 88,89%" + ], "cell_type": "markdown", - "metadata": {}, + "metadata": {} + }, + { + "source": [ + "\n" + ], + "cell_type": "markdown", + "metadata": {} + }, + { "source": [ - "### In order to solve and test the Mirmin Peres Magic Square (MPG) game on a quantum computer, we \n", - "employ the Qiskit framework and the IBM Q-Experience. Subsequently, this notebook will introduce the quantum circuit solving the MPG game, quantify it's performance on both simulated and real devices, and finally provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game.\n", + "However, in order to solve this game with a probility of 100%, one might look in the direction of quantum mechanics, quantum computing and quantum pseudo-telepathy where superposition and entanglement have already solved problems unfeasible on classical systems. Then, to realise this feat the Python Qiskit framework was employed, [extended from this Medium atricle] (https://medium.com/qiskit/this-proof-demonstrates-a-quantum-advantage-even-for-noisy-quantum-computers-b44a738801ad). Subsequently, this notebook will introduce the quantum circuit solving the MPG game, provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game and quantify it's performance on both simulated and real quantum computers.\n", "\n", - "#### For the subsequent circuit definition, evaluation and illustration, we first have to define auxiliary functions and import the corresponding modules from Qiskit and Numpy" - ] + "\n", + "First, for the subsequent circuit definition, evaluation and illustration, we first have to define auxiliary functions and import the corresponding modules from Qiskit and Numpy" + ], + "cell_type": "markdown", + "metadata": {} }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "from qiskit import *\n", "import numpy as np\n", "from qiskit.test.mock import *\n", - "from qiskit.visualization import plot_state_qsphere" + "from qiskit.visualization import plot_state_qsphere\n" ] }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 3, "metadata": {}, "outputs": [], "source": [ @@ -88,7 +102,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 4, "metadata": {}, "outputs": [], "source": [ @@ -128,7 +142,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 5, "metadata": {}, "outputs": [], "source": [ @@ -144,7 +158,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 6, "metadata": {}, "outputs": [], "source": [ @@ -189,37 +203,37 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 8, "metadata": {}, "outputs": [ { - "name": "stdout", "output_type": "stream", + "name": "stdout", "text": [ "Alice and Bob, select a column or row value of 1, 2, or 3\n", "Alice choice (column):\n", "Bob choice (row):\n", - " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", - "q163_0: ┤ H ├──■────────░─┤ H ├─░─┤M├─────────\n", - " ├───┤ │ ░ ├───┤ ░ └╥┘┌─┐ \n", - "q163_1: ┤ H ├──┼────■───░─┤ I ├─░──╫─┤M├──────\n", - " └───┘┌─┴─┐ │ ░ ├───┤ ░ ║ └╥┘┌─┐ \n", - "q164_0: ─────┤ X ├──┼───░─┤ H ├─░──╫──╫─┤M├───\n", - " └───┘┌─┴─┐ ░ ├───┤ ░ ║ ║ └╥┘┌─┐\n", - "q164_1: ──────────┤ X ├─░─┤ H ├─░──╫──╫──╫─┤M├\n", - " └───┘ ░ └───┘ ░ ║ ║ ║ └╥┘\n", - " c20: 1/═══════════════════════════╩══╬══╬══╬═\n", - " 0 ║ ║ ║ \n", - " ║ ║ ║ \n", - " c21: 1/══════════════════════════════╩══╬══╬═\n", - " 0 ║ ║ \n", - " ║ ║ \n", - " c22: 1/═════════════════════════════════╩══╬═\n", - " 0 ║ \n", - " ║ \n", - " c23: 1/════════════════════════════════════╩═\n", - " 0 \n", - "[('fake_yorktown', '51.86767578125%'), ('fake_melbourne', '91.73583984375%'), ('fake_vigo', '95.41015625%'), ('fake_ourense', '94.27490234375%'), ('fake_valencia', '91.14990234375%'), ('fake_london', '92.05322265625%'), ('fake_burlington', '91.5771484375%'), ('fake_essex', '93.603515625%'), ('qasm_simulator', '100.0%')]\n" + " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", + "q0_0: ┤ H ├──■────────░─┤ H ├─░─┤M├─────────\n", + " ├───┤ │ ░ ├───┤ ░ └╥┘┌─┐ \n", + "q0_1: ┤ H ├──┼────■───░─┤ I ├─░──╫─┤M├──────\n", + " └───┘┌─┴─┐ │ ░ ├───┤ ░ ║ └╥┘┌─┐ \n", + "q1_0: ─────┤ X ├──┼───░─┤ H ├─░──╫──╫─┤M├───\n", + " └───┘┌─┴─┐ ░ ├───┤ ░ ║ ║ └╥┘┌─┐\n", + "q1_1: ──────────┤ X ├─░─┤ H ├─░──╫──╫──╫─┤M├\n", + " └───┘ ░ └───┘ ░ ║ ║ ║ └╥┘\n", + "c0: 1/═══════════════════════════╩══╬══╬══╬═\n", + " 0 ║ ║ ║ \n", + " ║ ║ ║ \n", + "c1: 1/══════════════════════════════╩══╬══╬═\n", + " 0 ║ ║ \n", + " ║ ║ \n", + "c2: 1/═════════════════════════════════╩══╬═\n", + " 0 ║ \n", + " ║ \n", + "c3: 1/════════════════════════════════════╩═\n", + " 0 \n", + "100.0%\n" ] } ], @@ -260,76 +274,248 @@ " magicsquare_circuit.draw(output='mpl')\n", " print(magicsquare_circuit) \n", "\n", - " results_backends = []\n", + " job = execute(magicsquare_circuit, BasicAer.get_backend('qasm_simulator'), optimization_level=3, shots=8192) # run the job simulation\n", + " results = job.result().get_counts()\n", + " expected_states = check_result(results, alpha, beta)\n", + " state_accuracy = get_state_accuracy(results,expected_states,8192)\n", + " print(state_accuracy)\n", + "\n", + "else:\n", + " print(\"please enter values of 1, 2, or 3\")" + ] + }, + { + "source": [ + "#### Indeed, using an ideal quantum simulator, a winning probability of 100% is achieved, which exceeds the highest classical winning probability of 88.99% Beyond that, however, it is desirable to quantify the results on a real IBM quantum computer, which will be realized in the last section." + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Mathmatical background of the computation\n", + "\n", + "As we have demonstrated above, the quantum circuit solves the Mirmin-Peres game (MPG) on real quantum computers with a higher probability than with any classical strategy on ordinary computers. To shed some light on how and why this works, we present and explain the mathematical operations that lead to the solution of this game. In particular, we will first present a high-level equation describing the mathematical rationale and then we will take a look at each element of this equation explain it step by step.\n", "\n", - " backends = [FakeYorktown(), FakeMelbourne(), FakeVigo(), FakeOurense(), FakeValencia(), FakeLondon(),\n", - " FakeBurlington(), FakeEssex(), BasicAer.get_backend('qasm_simulator')]\n", "\n", - " for backend in backends:\n", - " job = execute(magicsquare_circuit, backend,optimization_level=3, shots=8192) # run the job simulation\n", - " results = job.result().get_counts()\n", - " expected_states = check_result(results, alpha, beta)\n", - " state_accuracy = get_state_accuracy(results,expected_states,8192)\n", - " results_backends.append((backend.name(), state_accuracy)) # store results\n", + "Recall that the quantum circuit has to play different moves for each different row/column instances of the MPG being selected, since there is no classical strategy that can win the game with probability = 1. Thus, to win with probability 1, the quantum circuit dynamically adapts to the input of each column and row played. Nevertheless, there is indeed one part of the circuit that remains static and is responsible for generating a matching qubit state and we will start of by focusing on that part. \n", "\n", - " print(results_backends)\n", + "The following Pauli operators govern what the ideal mathmatical operations, with the winning criteria in mind, are in correspondace to the selected row and column. \n", "\n", - "else:\n", - " print(\"please enter values of 1, 2, or 3\")" + "![title](Mirmin-Peres-Game/Pauli_Operators.png)\n", + "\n", + "\n", + "These operators can then be translated into the following quantum gates for the respective row / column selected. To further understand why these Pauli operators are translated to the gates, see.: [Quantum advantage in noisy flat circuits](https://arxiv.org/pdf/1904.01502.pdf) \n", + "\n", + "![title](Mirmin-Peres-Game/Circuit_Operations.png)\n", + "\n", + "\n", + "\n", + "\n", + "Abstractly, then, as evident from the gates and the circuit above, see (a) & (b), the elements can be described by the following high-level formula:\n", + "\n", + "$$ Quantum Solution MPG = Initial Vector * Dynamic Matrix(row, column)$$\n", + "\n", + "#### Starting with the first part of the equation, the intial vector, of the MPG circuit is defined by quantum gates as follows:\n" + ] + }, + { + "cell_type": "code", + "execution_count": 51, + "metadata": {}, + "outputs": [ + { + "output_type": "execute_result", + "data": { + "text/plain": [ + "
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\n" + }, + "metadata": {}, + "execution_count": 51 + } + ], + "source": [ + "q = QuantumRegister(4) \n", + "c = ClassicalRegister(4) \n", + "qc = QuantumCircuit(q, c) \n", + "qc.h(q[0])\n", + "qc.h(q[1])\n", + "qc.cx(q[0],q[2])\n", + "qc.cx(q[1],q[3])\n", + "#qc.barrier()\n", + "qc.draw(output='mpl')\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### The very starting vector state of the MPG circuit, prior to any quantum gate interaction, corresponds to the Dirac bracket |0000> and is hence described by the following vector: \n", + "$$ InitialQubits= \\frac 1 {\\sqrt{2}} {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}$$" ] }, + { + "source": [ + "#### In terms of the gates, these are the mathmatical represetations of the relevant gates illustrated above:\n", + "\n", + "$$\n", + "I = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} \n", + "$$\n", + "\n", + "$$\n", + "H = \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix}\n", + "$$\n", + "\n", + "$$\n", + "CNOT_{(i, i+2)} = \\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ \n", + "0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\ \n", + "0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ \n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ \n", + "0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\ \n", + "0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ \n", + "0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ \n", + "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\ \n", + "\\end{pmatrix}\n", + "$$" + ], + "cell_type": "markdown", + "metadata": {} + }, { "cell_type": "markdown", "metadata": {}, "source": [ - "# Quantum Circuit of Mermim-Peres-Game\n", + "#### While the static matrix, represented as the quantum gates in the illustration above, is created by the appliance of two CNOT gates to the first and third qubit and the second and fourth qubit, as well as two Hadamard gates applied to the first and second qubit. Mathematically, this is described by the following equation: \n", "\n", - "#### Now the circuit modeling the Mirmim Peres game can be created and the user may select the respective column and row that ought to be played. Subsequently, the probability of winning with respect to the game moves of the circuit are shown and evaluated." + "$$ Initial Vector = (I ⨂ CNOT_{0|2}) ⨂ (CNOT_{1|3} ⨂ I) * (I ⨂ I ⨂ H ⨂ H) * InitialQubits $$\n", + "\n", + "Subsequently inserting the corresponding matrices into this equation results in the following matrix:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "#### Indeed, using an ideal quantum simulator, a winning probability of 1 can be achieved, exceeding the highest classical-winning probability of 8/9. Beyond that, however, it is desirable to quantify the results on a real IBM quantum computer, which will be realized in the last section. " + "$$\n", + "Inital Vector = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} *\n", + "\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} \n", + "*\n", + "\\frac 1 {\\sqrt{2}} {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", + "$$\n", + "\n", + "$$ \n", + "Inital Vector = \\begin{bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\ ⨂ \\begin {bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", + "*\n", + "\\begin{pmatrix} 1 & 0 & 0 & 0\\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0\\\\ 0 & 0 & 0 & 1 \\end{pmatrix} \n", + "\n", + "⨂ \n", + "\n", + "\\frac 1 {2} \\begin{pmatrix} 1 & 1 & 1 & 1\\\\ 1 & -1 & 1 & -1 \\\\ 1 & 1 & -1 & -1\\\\ 1 & -1 & -1 & 1 \\end{pmatrix}\n", + "*\n", + "\\frac 1 {\\sqrt{2}} {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", + "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "# Mathmatical background of the computation\n", + "$$ \n", + "InitialVector = \\begin{bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", + "*\n", + "\\begin{bmatrix}\\ \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} \\\\\\end{bmatrix}\\\n", + "*\n", + "\\frac 1 {\\sqrt{2}} {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", + "$$\n", + "\n", + "$$\n", + "=\\begin{bmatrix}\\ \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2}& 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} \\\\\\ \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", + "*\n", + "\\frac 1 {\\sqrt{2}} {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", + "$$\n", + "$$\n", + "= \\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\ 0\\\\ \\frac 1 {2}\\end{pmatrix}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### This vector holds the relevant qubits that can be further considered and modified in the dynmaic part of the circuit so that valid winning can be created, these Qubit states are:\n", + "$$= \\frac 1 {2} (|0000\\rangle+|0101\\rangle+|1010\\rangle+|1111\\rangle)$$\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Furthermore, when analyzing the initial vector, it becomes apparent that when multiplying any matrix by it, only four relevenate cells per row in the multiplier remain open for consideration, since the others are all multiplied by 0 and thereby likewise drop of with 0% probability as a result. Thus, we can eliminate the majority of the entries whatsoever. As an example we have prepared the matrix below, and denoted all entries with an i, that have to be computed and all others that result in any case with 0 with -.\n", "\n", - "#### As we have demonstrated above, the quantum circuit solve the Mirmin-Peres game (MPG) on real quantum computers with a higher probability than with any classical strategy on ordinary computers. To shed some light on how and why this works, we present and explain the mathematical operations that lead to the solution of this game. In particular, we will first present a high-level equation describing the mathematical rationale, and we will then take a look at each element of this equation and finally explain it in one go using a concrete example.\n", + "$$ = \n", + "\\begin{bmatrix}\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\end{bmatrix} \n", + "*\n", + "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\ \\frac 1 {2}\\end{pmatrix}\n", + "$$\n", "\n", - "#### Recall that the quantum circuit has to play different moves for each different row/column instances of the MPG being selected, since there is no classical strategy that can win the game with probability = 1. Thus, to win with probability 1, the quantum circuit dynamically adapts to the input of each column and row played. Nevertheless, there is indeed one part of the circuit that remains static and is responsible for generating a matching qubit state and we will start of by focusing on that part. Abstractly, then, the quantum circuit can be described by the following formula: (For a technical background on why the gates were applied, see: [Quantum advantage in noisy flat circuits](https://www.nature.com/articles/s41567-020-0948-z)\n", + "Let us now take a look at an concrete example, which show and proof that only the i-rows are relevant for the subsequent computation:\n", "\n", + "$$\n", + "Winnning Plays= \n", + "\\begin{bmatrix}\\ \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \\end{bmatrix} * \n", + "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\ \\frac 1 {2}\\end{pmatrix}\n", + "$$\n", "\n", - "$$ Quantum Solution MPG = Initial Vector * Dynamic Matrix(row, column)$$\n", + "Multiplying each row element with each coloumn element yields the following results:\n", "\n", + "$$\n", + "= \\frac 1 {\\sqrt{8}}\n", + "$$\n", "\n", - "#### Starting with the first part of the equation, the intial vector, of the MPG circuit is defined by quantum gates as follows:" + "\n", + "However, considering only the i-row entries yields precisely the same results:\n", + "$$\n", + "Winnning Plays= \n", + "\\begin{bmatrix}\\ \\frac 1 {\\sqrt{8}} & - & - & - & - & \\frac 1 {\\sqrt{8}}& - & - & - & - & 0 & - & - & - & - & 0 \\end{bmatrix} * \n", + "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\ \\frac 1 {2}\\end{pmatrix}\n", + "$$\n", + "$$\n", + "= \\frac 1 {\\sqrt{8}}\n", + "$$\n", + "#### And indeed, this idea is very powerful because it lets us visually see what outcomes may emerge as well asa better idea of which outcomes may fall off and which may be a win in the MPG. As long as postive result emerges the MPG is won, no matter the probability of that play occuring" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Now, in order to consider the later, dynamic, part of the equation we will have to select an example. In this case we have selected row: 1 and column: 1. The corresponding quantum circuit is as follows:" ] }, { "cell_type": "code", - "execution_count": 403, + "execution_count": 439, "metadata": {}, "outputs": [ { "data": { - "image/png": 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\n", + "image/png": 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" ] }, - "execution_count": 403, + "execution_count": 439, "metadata": {}, "output_type": "execute_result" } ], "source": [ "q = QuantumRegister(4) \n", - "c = ClassicalRegister(3) \n", + "c = ClassicalRegister(4) \n", "qc = QuantumCircuit(q, c) \n", - "qc.h(q[0])\n", - "qc.h(q[1])\n", - "qc.cx(q[1],q[3])\n", - "qc.cx(q[0],q[2])\n", + "#qc.h(q[0])\n", + "#qc.h(q[1])\n", + "#qc.cx(q[1],q[3])\n", + "#qc.cx(q[0],q[2])\n", "qc.barrier()\n", + "qc.h(q[0])\n", + "qc.id(q[1])\n", + "qc.h(q[2])\n", + "qc.h(q[3])\n", "qc.draw(output='mpl')" ] }, @@ -878,44 +990,43 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "#### The very starting vector state of the MPG circuit, prior to any quantum gate interaction, corresponds to the Dirac bracket |0000> and is hence described by the following vector: \n", - "$$ = {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$ \n", - "StaticMatrix = \\begin{bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\ ⨂ \\begin {bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", - "*\n", - "\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} \n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### While the static matrix, represented as the quantum gates in the illustration above, is created by the appliance of two CNOT gates to the first and third qubit and the second and fourth qubit, as well as two Hadamard gates applied to the first and second qubit. Mathematically, this is described by the following equation: \n", - "\n", - "$$ Static Matrix= CNOT_{2|4} ⨂ CNOT_{1|3} * (I ⨂ I ⨂ H ⨂ H) $$\n", - "\n", - "Subsequently inserting the corresponding matrices into this equation results in the following matrix:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$ \n", - "StaticMatrix = \\begin{bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", - "*\n", - "\\begin{bmatrix}\\ \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} \\\\\\end{bmatrix}\\\n", + "#### The translation of these circuit gate operations is in turn equivalent to the following mathematical matrice multiplications:\n", + "$$ DynamicMatrix(1, 1) = H ⨂ H ⨂ I ⨂ H $$\n", "$$\n", - "\n", + "= \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix}\n", "$$\n", - "=\\begin{bmatrix}\\ \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2}& 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} \\\\\\ \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", + "$$\n", + "=\n", + "\\begin{bmatrix}\\ \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", + "\\\\\\\n", + " 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", + "\\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", + "\\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", + "\\\\\\\n", + " \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0\n", + " \\\\\\\n", + " \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0\n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}\n", + " \\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0\n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \\\\\\end{bmatrix}\\ \n", "$$" ] }, @@ -923,1419 +1034,82 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "#### The matrix above is the final static matrix for each instance played with the MPG quantum circuit. This matrix can be further reduced by multplication with the, likewise static, inital state vector. Furthermore, this presents us with the Qubits considered for further usage in the dynmic parts of the circuit" + "#### Now, as we have both, the inital vector and the dynamic matrix, we can compute the winning plays for Alice and Bob with selected column and row of 1. In prinicple we can apply the rule of the i-rows, that we have established above, to this matrix-vector multiplication, however, in this case for the sake of completness we let all entries remain. But feel free to check the rule!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "$$ = Static Matrix * Inital State $$\n", - "$$ \n", - "= \\begin{bmatrix}\\ \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2}& 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} \\\\\\ \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", - "*\n", - "\\frac 1 {\\sqrt{2}} {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", "$$\n", - "\n", - "$$ \n", - "= \n", - "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\ 0\\\\ \\frac 1 {2}\\end{pmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### Then this vector holds the relevant qubits that can be further considered and modified / computed with in the dynmaic part of the circuit, these are:\n", - "$$= \\frac 1 {2} (|0000\\rangle+|0101\\rangle+|1010\\rangle+|1111\\rangle)$$\n", - "#### Moreover, due to the nature of matrix multiplication, this vector allows the following simplification. For each row of the first matrix, we only need to consider the numbers marked in red and calculate their product sum with the static vector on the right, as the other positions are multiplied by zero." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$ = \n", - "\\begin{bmatrix}\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\ \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} & 0 & 0 & 0 & 0 & \\color{red}{0} \\\\\\end{bmatrix} \n", - "*\n", - "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\ 0\\\\ \\frac 1 {2}\\end{pmatrix}\n", - "$$" - ] - }, - { - "cell_type": "code", - "execution_count": 439, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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- "text/plain": [ - "
" - ] - }, - "execution_count": 439, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "q = QuantumRegister(4) \n", - "c = ClassicalRegister(3) \n", - "qc = QuantumCircuit(q, c) \n", - "#qc.h(q[0])\n", - "#qc.h(q[1])\n", - "#qc.cx(q[1],q[3])\n", - "#qc.cx(q[0],q[2])\n", - "qc.barrier()\n", - "qc.h(q[0])\n", - "qc.id(q[1])\n", - "qc.h(q[2])\n", - "qc.h(q[3])\n", - "qc.draw(output='mpl')" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### Now, in order to consider the later, dynamic, part of the equation we will have to select an example. In this case we have selected row: 1 and column: 1. The corresponding quantum circuit is as follows:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### The translation of these circuit gates into mathematical matrices is in turn equivalent to the following sequence of operations:\n", - "$$ = H ⨂ H ⨂ I ⨂ H $$\n", - "$$\n", - "= \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix}\n", - "$$\n", - "$$\n", - "DynamicMatrix(1, 1) =\n", - "\\begin{bmatrix}\\ \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \\\\\\ \n", - "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 \n", - "\\\\\\ \n", - "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", - "\\\\\\\n", - " 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", - "\\\\\\ \n", - "\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 \n", - "\\\\\\ \n", - "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \n", - "\\\\\\ \n", - "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", - "\\\\\\ \n", - "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", - "\\\\\\\n", - " \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0\n", - " \\\\\\\n", - " \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0\n", - " \\\\\\ \n", - "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", - " \\\\\\ \n", - "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}\n", - " \\\\\\ \n", - "\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \n", - "\\\\\\ \n", - "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0\n", - " \\\\\\ \n", - "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", - " \\\\\\ \n", - "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \\\\\\end{bmatrix}\\ \n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### The matrix above is then the one employed when row / column: 1 is selected. In order to now ascertain the winning combinations for that particular game, we have to multiply the dynamic matrix with the static." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " $$\n", - "Winning Plays= \\begin{bmatrix}\\ \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \\\\\\ \n", - "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 \n", - "\\\\\\ \n", - "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", - "\\\\\\\n", - " 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", - "\\\\\\ \n", - "\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 \n", - "\\\\\\ \n", - "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \n", - "\\\\\\ \n", - "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", - "\\\\\\ \n", - "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", - "\\\\\\\n", - " \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0\n", - " \\\\\\\n", - " \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0\n", - " \\\\\\ \n", - "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", - " \\\\\\ \n", - "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}\n", - " \\\\\\ \n", - "\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \n", - "\\\\\\ \n", - "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0\n", - " \\\\\\ \n", - "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", - " \\\\\\ \n", - "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \\\\\\end{bmatrix}\\ \n", - "*\n", - "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\ 0\\\\ \\frac 1 {2}\\end{pmatrix}\n", - "$$\n", - "$$\n", - "= \\begin{pmatrix} \\frac 1 {\\sqrt{8}} \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\ -\\frac 1 {\\sqrt{8}} \\\\0 \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\ 0\\\\ -\\frac 1 {\\sqrt{8}} \\end{pmatrix}\n", - "$$\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### Finally, this vector contains the qubits that can be played to win the game with probability 1, these are:\n", - "\n", - "$$= \\frac 1 {8} (|0000\\rangle+|0100\\rangle+|0101\\rangle+|01111\\rangle+|1000\\rangle)+|1011\\rangle)+|1101\\rangle)+|1111\\rangle)$$\n", - "\n", - "#### To validate these mathematical results, we will run the quantum circuit in one go and compare the respective qubit results" - ] - }, - { - "cell_type": "code", - "execution_count": 460, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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\n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - "\n" - ], + "Winning Plays = DynamicMatrix * InitialVector\n", + "$$\n", + "$$\n", + "= \\begin{bmatrix}\\ \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", + "\\\\\\\n", + " 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", + "\\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", + "\\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", + "\\\\\\\n", + " \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0\n", + " \\\\\\\n", + " \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0\n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}\n", + " \\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 \n", + "\\\\\\ \n", + "\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}}& 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0\n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} \n", + " \\\\\\ \n", + "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \\\\\\end{bmatrix}\\ \n", + "*\n", + "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\ 0\\\\ \\frac 1 {2}\\end{pmatrix}\n", + "$$\n", + "$$\n", + "= \\begin{pmatrix} \\frac 1 {\\sqrt{8}} \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\ -\\frac 1 {\\sqrt{8}} \\\\0 \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\ 0\\\\ -\\frac 1 {\\sqrt{8}} \\end{pmatrix}\n", + "$$\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Finally, this vector contains the qubits that can be played to win the game with probability 1, these are:\n", + "\n", + "$$= \\frac 1 {8} (|0000\\rangle+|0100\\rangle+|0101\\rangle+|01111\\rangle+|1000\\rangle)+|1011\\rangle)+|1101\\rangle)+|1111\\rangle)$$\n", + "\n", + "#### To validate these mathematical results, we will run the quantum circuit in one go and compare the respective qubit results" + ] + }, + { + "cell_type": "code", + "execution_count": 52, + "metadata": {}, + "outputs": [ + { + "output_type": "execute_result", + "data": { "text/plain": [ - "
" - ] + "
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\n" }, - "execution_count": 460, "metadata": {}, - "output_type": "execute_result" + "execution_count": 52 } ], "source": [ @@ -2347,10 +1121,8 @@ "qc.cx(q[1],q[3])\n", "qc.cx(q[0],q[2])\n", "qc.barrier()\n", - "qc.h(q[0])\n", - "qc.id(q[1])\n", - "qc.h(q[2])\n", - "qc.h(q[3])\n", + "U(qc,1,0,1) \n", + "V(qc,1,2,3)\n", "qc.measure(0,0)\n", "qc.measure(1,1)\n", "qc.measure(2,2)\n", @@ -2388,15 +1160,66 @@ "#### Evidently, the results of the mathematical calculation match those of the circuit and a win rate of 100% is achieved!" ] }, + { + "source": [ + "------------------------------------------------------------------------------------------------------------------------------------------------------------------\n", + "To reproduce these operations for other configurations, it is recommended to use the Unitary simulator - follow this simple guide:\n", + "\n", + "1. Install \"pip install git+https://github.com/qiskit-community/qiskit-textbook.git#subdirectory=qiskit-textbook-src\"\n", + "\n", + "2. Create new Cell & Fill:\n", + "from qiskit_textbook.tools import array_to_latex;\n", + "result = execute(qc, Aer.get_backend('unitary_simulator')).result().get_unitary(qc);\n", + "array_to_latex(result, pretext=\"\\\\text{Circuit = }\\n\");\n", + "------------------------------------------------------------------------------------------------------------------------------------------------------------------" + ], + "cell_type": "markdown", + "metadata": {} + }, { "cell_type": "markdown", "metadata": {}, "source": [ - "# Circuit Performance on Real Quantum Devices\n", + "# Circuit Performance on Real Quantum Devices" + ] + }, + { + "source": [ + "#### Thus far all executions of the Quantum circuits were performed on ideal quantum simulators, rather than actual quantum hardware. Real quantum computers have noise, decoherence and other periphery hardware issues, which can reduce the performance and fidelity of the executions. Hence, in order to ascertain the actucal performance and not only the theoretical we will execute our circuit on real quantum computers as well as simulators of real computers that include the above mentioned errors alike. \n", "\n", - "#### Here we evaluate the performance of the ibmq_athens. Please note that you will have to add your own API token from the IBM Quantum website (https://quantum-computing.ibm.com)" + "#### We start off with the non-ideal simulators of the real quantum computers, so called FakeDevices. The use of FakeVigo, simualating the quantum Computer Vigo yields the following results." + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "95.27587890625%\n" + ] + } + ], + "source": [ + "job = execute(magicsquare_circuit, FakeVigo(), optimization_level=3, shots=8192) # run the job simulation\n", + "results = job.result().get_counts()\n", + "expected_states = check_result(results, alpha, beta)\n", + "state_accuracy = get_state_accuracy(results,expected_states,8192)\n", + "print(state_accuracy)" ] }, + { + "source": [ + "#### Now in order to evaluate the performance of real devices we select the ibmq_athens computer to benchmark the MPG. Please note that you will have to add your own API token from the IBM Quantum website (https://quantum-computing.ibm.com) so that this cell executes" + ], + "cell_type": "markdown", + "metadata": {} + }, { "cell_type": "code", "execution_count": 53, @@ -2502,7 +1325,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "#### Evidently, the MPG circuit outperforms the classical circuit by about 7.8% (88.89% vs. 96.69%) on a real, noisy, quantum computer." + "#### As shown above, the MPG circuit outperforms the classical circuit by about 7.8% (88.89% vs. 96.69%) on a real, noisy, quantum computer." ] } ], @@ -2522,9 +1345,9 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.5" + "version": "3.8.5-final" } }, "nbformat": 4, "nbformat_minor": 2 -} +} \ No newline at end of file diff --git a/Mirmin-Peres-Game/Circuit_Operations.png b/Mirmin-Peres-Game/Circuit_Operations.png new file mode 100644 index 0000000000000000000000000000000000000000..493f1c3e0c7b8cb47b596cffaf840addcca2da1d GIT binary patch literal 77515 zcmeEuWmuKl8ZEIvx*HXwq*DO_2?1%OyHmPL8U&P7x;v!1kuGULIwhsM>(1qlbN0E< zIX~~OyPw75g0;T2<~Qej-*>!YjJZF`$%>()kf6Z8z@STrizvXrAUMOo!1f~{g1=af zwMv6ugv^A6r>AOw25eVPI%K8|vxdh%?gm>g((2^$s%7q1d}9eE1Nmpy%D! zIzZmv+D1N@o*1vCwM2xq1h3Q%lO^9)Yk?L@epSKTTeNy9#>u|=t+}D8Zq8P!=^jn` zl-7rVx|*9<{2q2Wd(|%!2NAIwLatfglnv8q4O<}^>=O=GA!PC~TX@PsaEA)%8b(b< z{1e9~PlR3q@gZ?(*orimX&wpju zq{?S5A)#d`Aq+Pey?lf~2~R1Eh$!bOtmI7qA4HioMfY9}g_SXs3m$>%NvAzNG0_yGh$$i=>7eD z>h=BoA~hP?02eV41|A#?J)L&staWuXm{S#FH3<_L85nx-90>+C$P5Mop231267T~9 z10NX(^8oyd1Aau(;r@CR!8slNujjD+&^HPx3QI_Ue-#bwjg74x-r6{h_k>7+kD4@7 zQgc+3k>)Y9v0~OYvN14bcD1sFUIN4C$^)KS89VBeyINUVJMg%^r1<>~9`GFcXBG
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zPf1tu^&SAsVFK97xJi`9A&8jzd-0%Q*R1C3ngEXK^e@zmqyj3C!&U}JYg&K%zhHf@ z5-imhcd~T4LUBZis78NB3B+SQ;F9|}%UUaD+CI;t_b(BU#tV8tY?liVC!P-QKb0C? zL~D;Qf>)O|wt&X1FTUUf$N&~`Xe(36C;f!CRtwTve$xYy6By(3-E+I_>m5uksHbHH zSXFA-xRs+2lA#(DJ9K*<=RzZD>eT?G9Y#3ysDI-lfiE%re}n}>6(qN(Dr5EU<_Wn? z3Ar~|cPg{4%2lk?1n==Aaj4%)GJ{pqx#7bGACFoTp*9(}pBI@mdw9R)K>o`-By-`+ z+6z-!*QN>@!mRz1_PSV*5h_eB)4J@2Y~1slh3SaRx!iVxwEie`wx$5r=L(5&v!e7k z630V3|7(W4Hu&VY|FI7+ETd~HiV4dX{$o1|}E zm;c5n3&s(F6Z_}g1~9xraB0=V4)2x!<)+L*<>ejuUhOZ5 u^Z~p`gCi@rK>zOu1kAtX|L4 Date: Tue, 30 Mar 2021 13:37:02 +0200 Subject: [PATCH 17/30] fix: split removed from check winning --- "Mermin\342\200\223Peres-Game.ipynb" | 91 ++++++++++++---------------- 1 file changed, 39 insertions(+), 52 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index b42b8f9..778df79 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -33,7 +33,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 1, "metadata": {}, "outputs": [], "source": [ @@ -45,7 +45,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 212, "metadata": {}, "outputs": [], "source": [ @@ -63,8 +63,10 @@ "\n", " #print(\"Quantum circuit solution\")\n", " #print(i+\"\\n\")\n", + " \n", " i_orginal = i\n", - " i = np.array(list(i.split())).astype(int)[::-1]\n", + " i = np.array(list(i.replace(\" \", \"\")), dtype=np.int32)[::-1]\n", + " \n", " # bring array in correct i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", " a = i[0:2]\n", " a_sum = np.sum(a)\n", @@ -102,7 +104,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 213, "metadata": {}, "outputs": [], "source": [ @@ -142,7 +144,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 214, "metadata": {}, "outputs": [], "source": [ @@ -158,7 +160,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 215, "metadata": {}, "outputs": [], "source": [ @@ -203,7 +205,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 211, "metadata": {}, "outputs": [ { @@ -213,26 +215,26 @@ "Alice and Bob, select a column or row value of 1, 2, or 3\n", "Alice choice (column):\n", "Bob choice (row):\n", - " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", - "q0_0: ┤ H ├──■────────░─┤ H ├─░─┤M├─────────\n", - " ├───┤ │ ░ ├───┤ ░ └╥┘┌─┐ \n", - "q0_1: ┤ H ├──┼────■───░─┤ I ├─░──╫─┤M├──────\n", - " └───┘┌─┴─┐ │ ░ ├───┤ ░ ║ └╥┘┌─┐ \n", - "q1_0: ─────┤ X ├──┼───░─┤ H ├─░──╫──╫─┤M├───\n", - " └───┘┌─┴─┐ ░ ├───┤ ░ ║ ║ └╥┘┌─┐\n", - "q1_1: ──────────┤ X ├─░─┤ H ├─░──╫──╫──╫─┤M├\n", - " └───┘ ░ └───┘ ░ ║ ║ ║ └╥┘\n", - "c0: 1/═══════════════════════════╩══╬══╬══╬═\n", - " 0 ║ ║ ║ \n", - " ║ ║ ║ \n", - "c1: 1/══════════════════════════════╩══╬══╬═\n", - " 0 ║ ║ \n", - " ║ ║ \n", - "c2: 1/═════════════════════════════════╩══╬═\n", - " 0 ║ \n", - " ║ \n", - "c3: 1/════════════════════════════════════╩═\n", - " 0 \n", + " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", + "q1120_0: ┤ H ├──■────────░───X──┤ H ├─░─┤M├─────────\n", + " ├───┤ │ ░ │ ├───┤ ░ └╥┘┌─┐ \n", + "q1120_1: ┤ H ├──┼────■───░───X──┤ I ├─░──╫─┤M├──────\n", + " └───┘┌─┴─┐ │ ░ ┌───┐└───┘ ░ ║ └╥┘┌─┐ \n", + "q1121_0: ─────┤ X ├──┼───░─┤ H ├──────░──╫──╫─┤M├───\n", + " └───┘┌─┴─┐ ░ ├───┤ ░ ║ ║ └╥┘┌─┐\n", + "q1121_1: ──────────┤ X ├─░─┤ H ├──────░──╫──╫──╫─┤M├\n", + " └───┘ ░ └───┘ ░ ║ ║ ║ └╥┘\n", + " c159: 1/════════════════════════════════╩══╬══╬══╬═\n", + " 0 ║ ║ ║ \n", + " ║ ║ ║ \n", + " c160: 1/═══════════════════════════════════╩══╬══╬═\n", + " 0 ║ ║ \n", + " ║ ║ \n", + " c161: 1/══════════════════════════════════════╩══╬═\n", + " 0 ║ \n", + " ║ \n", + " c162: 1/═════════════════════════════════════════╩═\n", + " 0 \n", "100.0%\n" ] } @@ -323,22 +325,9 @@ }, { "cell_type": "code", - "execution_count": 51, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "output_type": "execute_result", - "data": { - "text/plain": [ - "

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\n" - }, - "metadata": {}, - "execution_count": 51 - } - ], + "outputs": [], "source": [ "q = QuantumRegister(4) \n", "c = ClassicalRegister(4) \n", @@ -1096,20 +1085,20 @@ }, { "cell_type": "code", - "execution_count": 52, + "execution_count": 24, "metadata": {}, "outputs": [ { "output_type": "execute_result", "data": { "text/plain": [ - "
" + "
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\n" 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\n" }, "metadata": {}, - "execution_count": 52 + "execution_count": 24 } ], "source": [ @@ -1132,24 +1121,22 @@ }, { "cell_type": "code", - "execution_count": 457, + "execution_count": 216, "metadata": {}, "outputs": [ { - "name": "stdout", "output_type": "stream", + "name": "stdout", "text": [ - "{'0000': 1027, '0010': 1062, '0101': 1032, '0111': 993, '1000': 1047, '1010': 1046, '1101': 975, '1111': 1010}\n", "Accuracy: 100.0%\n" ] } ], "source": [ "backend = Aer.get_backend('qasm_simulator')\n", - "result = execute(qc, backend,shots=8192).result().get_counts()\n", + "results = execute(qc, backend, optimization_level=3, shots=8192).result().get_counts()\n", "expected_states = check_result(results, 1, 1)\n", "state_accuracy = get_state_accuracy(results,expected_states,8192)\n", - "print(result)\n", "print(\"Accuracy: \"+state_accuracy)" ] }, From 02a80605889fe96b8bb7c1358a67b023ae5c9997 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Thu, 1 Apr 2021 11:38:52 +0200 Subject: [PATCH 18/30] fix: function usage --- "Mermin\342\200\223Peres-Game.ipynb" | 550 +++------------------------ 1 file changed, 54 insertions(+), 496 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 778df79..7ade691 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -33,7 +33,7 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 2, "metadata": {}, "outputs": [], "source": [ @@ -45,7 +45,7 @@ }, { "cell_type": "code", - "execution_count": 212, + "execution_count": 3, "metadata": {}, "outputs": [], "source": [ @@ -104,7 +104,7 @@ }, { "cell_type": "code", - "execution_count": 213, + "execution_count": 4, "metadata": {}, "outputs": [], "source": [ @@ -144,7 +144,7 @@ }, { "cell_type": "code", - "execution_count": 214, + "execution_count": 5, "metadata": {}, "outputs": [], "source": [ @@ -160,7 +160,7 @@ }, { "cell_type": "code", - "execution_count": 215, + "execution_count": 6, "metadata": {}, "outputs": [], "source": [ @@ -205,7 +205,7 @@ }, { "cell_type": "code", - "execution_count": 211, + "execution_count": 7, "metadata": {}, "outputs": [ { @@ -215,26 +215,26 @@ "Alice and Bob, select a column or row value of 1, 2, or 3\n", "Alice choice (column):\n", "Bob choice (row):\n", - " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", - "q1120_0: ┤ H ├──■────────░───X──┤ H ├─░─┤M├─────────\n", - " ├───┤ │ ░ │ ├───┤ ░ └╥┘┌─┐ \n", - "q1120_1: ┤ H ├──┼────■───░───X──┤ I ├─░──╫─┤M├──────\n", - " └───┘┌─┴─┐ │ ░ ┌───┐└───┘ ░ ║ └╥┘┌─┐ \n", - "q1121_0: ─────┤ X ├──┼───░─┤ H ├──────░──╫──╫─┤M├───\n", - " └───┘┌─┴─┐ ░ ├───┤ ░ ║ ║ └╥┘┌─┐\n", - "q1121_1: ──────────┤ X ├─░─┤ H ├──────░──╫──╫──╫─┤M├\n", - " └───┘ ░ └───┘ ░ ║ ║ ║ └╥┘\n", - " c159: 1/════════════════════════════════╩══╬══╬══╬═\n", - " 0 ║ ║ ║ \n", - " ║ ║ ║ \n", - " c160: 1/═══════════════════════════════════╩══╬══╬═\n", - " 0 ║ ║ \n", - " ║ ║ \n", - " c161: 1/══════════════════════════════════════╩══╬═\n", - " 0 ║ \n", - " ║ \n", - " c162: 1/═════════════════════════════════════════╩═\n", - " 0 \n", + " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", + "q0_0: ┤ H ├──■────────░───■──┤ H ├──────░─┤M├─────────\n", + " ├───┤ │ ░ ┌─┴─┐├───┤ ░ └╥┘┌─┐ \n", + "q0_1: ┤ H ├──┼────■───░─┤ X ├┤ I ├──────░──╫─┤M├──────\n", + " └───┘┌─┴─┐ │ ░ ├───┤└───┘┌───┐ ░ ║ └╥┘┌─┐ \n", + "q1_0: ─────┤ X ├──┼───░─┤ Z ├──■──┤ H ├─░──╫──╫─┤M├───\n", + " └───┘┌─┴─┐ ░ ├───┤ │ ├───┤ ░ ║ ║ └╥┘┌─┐\n", + "q1_1: ──────────┤ X ├─░─┤ Z ├──■──┤ H ├─░──╫──╫──╫─┤M├\n", + " └───┘ ░ └───┘ └───┘ ░ ║ ║ ║ └╥┘\n", + "c0: 1/═════════════════════════════════════╩══╬══╬══╬═\n", + " 0 ║ ║ ║ \n", + " ║ ║ ║ \n", + "c1: 1/════════════════════════════════════════╩══╬══╬═\n", + " 0 ║ ║ \n", + " ║ ║ \n", + "c2: 1/═══════════════════════════════════════════╩══╬═\n", + " 0 ║ \n", + " ║ \n", + "c3: 1/══════════════════════════════════════════════╩═\n", + " 0 \n", "100.0%\n" ] } @@ -325,9 +325,22 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 8, "metadata": {}, - "outputs": [], + "outputs": [ + { + "output_type": "execute_result", + "data": { + "text/plain": [ + "
" + ], + "image/svg+xml": "\n\n\n\n \n \n \n \n 2021-03-30T13:37:49.484335\n image/svg+xml\n \n \n Matplotlib v3.3.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n", + "image/png": 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\n" + }, + "metadata": {}, + "execution_count": 8 + } + ], "source": [ "q = QuantumRegister(4) \n", "c = ClassicalRegister(4) \n", @@ -488,475 +501,20 @@ }, { "cell_type": "code", - "execution_count": 439, + "execution_count": 9, "metadata": {}, "outputs": [ { + "output_type": "execute_result", "data": { - "image/png": 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wDKP28/LFtFzLysriugSjYFDnSNrY2tqit7eX6zKMVkJCAoWJBQb/jkT0q39QhowMBYkQFlCQCGEBBcnE0Yex7KAgmbhTp05xXYJRMPhRO6Jf27Zt4+R6O/GECaO+T33ul4JEOKGPOYq4RF07QlhAQTJxmZmZXJdgFChIJs7b25vrEowCBcnEvXxvDPJqKEiEsICCRAgLKEgm7q233uK6BKNAQTJx1dXVXJdgFChIhLCAgkQICyhIJq7/tmJkZChIhLCAgmTihjNDBhkYXf1NOFFQA7Q8Hv39ujoA785iv10KEuFEy2Pg5o9cV8Ee6tqZuP5ZCcnIUJBMXGJiItclGAUKkomTSCRcl2AUKEgmrq2tjesSjAIFiRAWUJBMXP+EbGRkKEgmLj8/n+sSjMKYDdKFCxcQFxcHoVAIHo8Hd3d3LF++HLdv31Zbr7S0FGZmZlp/uru7VeudP38e8+bNg4uLC6ysrODi4oIFCxagvLxcY9/Nzc3YsGEDgoODwePxYGZmBqVSqfdj1oetW7dyXYJRGLNBOnjwIO7du4ctW7agqKgIu3btQmVlJQICArROPZmamory8nK1HysrK9XjDx8+hFgsxoEDB3Du3Dl88sknePDgASQSCcrKytTaunHjBnJzc2Fvbz/m73lw+vRprkswCmP2yobMzEw4OTmpLZs7dy6mTZuGrKwspKSozzMqEokGncVcKpVCKpWqLYuMjISTkxOOHj2K4OBg1XKJRILW1lYALyYTKy4uHunhkCHk7QrFVJ+3MTvqTzotH20G+Y7U19eH1NRUCIVCWFtbQywWQ6FQQCQSYfXq1QCgESLgxUTIAoEAzc3NrNRha2sLKysrjBs3Tm25ublBPm2EQwb5F7Fy5Urs3LkT8fHxKCoqQkxMDJYuXYpbt24NOpesUqlEW1sbfHx8tLZpYWEBBwcHREdHo6GhQWsbvb296Onpwd27d1WXz6xZs4adAzNACoWC6xKMgsF17XJycnD8+HGUlpaqzj/CwsJw5coVFBQUDBiknp4exMfHQyAQqN61AGDixIlISkpCaGgo7OzsUFdXB7lcjqCgIFRXV8PT01OtnZCQEFy+fBkAMGnSJBQVFcHX13fYx/Hs2TPEx8ejpKQEnZ2d8Pf3x6efforp06cPuy19unbtGpydnbkuY8wzuCDt2bMHERERGifxXl5esLS01PpHzTAMVq1ahaqqKhQWFsLR0VH1mJ+fH/z8/FS/SyQShIeHQywWQy6XIzs7W62t7OxstLe3o6WlBYcPH0ZERATOnDkz7Nm/nz9/Di8vL+zevRuTJ09GSkoKpFIprl69qtP2ZmZmw9qfNps2bRpynfT09CHXS09PH3EtL1si+xqvTw8d1jZVX+7GN2dT1Zb1dD/FVJ+3dW5DoSjFhvAwnddnGEan9QwqSM3NzVAqlVr/Y5uamuDt7a020tYvMTERJ06cQE5ODsLDw4fcj1AoRGBgICoqKjQeE4lEqn8vWrQIQUFB2LhxI2pra4d1LOPHj8ef/vSfE+D169dDJpOhu7sb1tbWw2qLvDB7kUzrYIMhMLggAcDkyZPVlnd1dUGhUOCdd97R2CYpKQlZWVnIzs7WGHUbDMMwQ77qm5ubY9asWThy5IjO7Q6krKwM7u7uOodI11fCwdTX1w+5Tnp6ulpXWJu0tLQR1/KyjBJuvo8UEhKKvF0jf25fZlCDDQKBAADQ2Niotnzv3r24f/8+/P391ZbLZDKkpaUhIyMDK1as0Hk/169fR2Vl5aDD4cCL865Lly7By8tL57a1efz4MdatW4fdu3ePqB192LFjB9clGAWDekfy8PDAjBkzIJfLwefz4erqiry8PJw9exYA1AYa9u3bB7lcDqlUioCAALVump2dneoastjYWHh4eMDf3x/29vaoq6tDcnIyeDweZDKZapuoqCjMnDkTYrEYfD4fTU1NOHToEJRKJQoKCjRq7b/7Tk1NDQCgpKQE9fX1GD9+PCIjI1XrdXV1YeHChZBKpXjvvfdYfLbYwcVsfcbIjGGjD8GixsZGxMfHo6qqCo6OjoiLi8OECRMgk8nw5MkT2NjYAABCQ0MHHLoNCQlBaWkpACA5ORknT57EnTt30NnZCWdnZ8yfPx/bt2+Hu7u7apuUlBTk5eXh5s2b6OjoAJ/PR1BQEDZv3ow5c+Zo7GOgbqGbm5vqyornz59j8eLFcHJyYqV7OFy6dO2mT58+5ITMb775JlslqXDVtfN0BtbPZ79dgwuSNsuWLUNtba3OI16G4oMPPkBbWxu+/PJLWFiM/ps/BUmTvoJkUF27gdTU1Ax5PmNo7t69i+PHj8Pa2hr29vaq5d999x2mTp3KXWFELww+SE+fPkVjYyPWrl3LdSnD4ubmxsrIm74N9/Mxop3BB8nW1ha9vb1cl2G0srKyuC7BKBjU8DcZfQkJCVyXYBQoSCauf3STjAwFiRAWUJAIYQEFycQN9RkS0Y3Bj9oR/Tp16hQnlwm5Ooz6LvW63zFxZQN5NYZ8ZYOxoa4dISygIBHCAgqSicvMzOS6BKNAQTJx3t7eXJdgFChIJm6s3ynWUFCQCGEBBcnEvfXWW1yXYBQoSCauurqa6xKMAgWJEBZQkAhhAQXJxPXfVoyMDAWJEBZQkExcdHQ01yUYBfoaBeFEQQ3Q8nj09+vqALw7i/12KUiEEy2PublBpL5Q187E9c9KSEaGgmTiEhMTuS7BKFCQTJxEIuG6BKNAQTJxbW1tXJdgFChIhLCAgmTi+idkIyNDQTJx+fn5XJdgFMZskC5cuIC4uDgIhULweDy4u7tj+fLluH37ttp6paWlMDMz0/rT3d2tWu/8+fOYN28eXFxcYGVlBRcXFyxYsADl5eUa+25ubsaGDRsQHBwMHo8HMzMzKJVKvR+zPmzdupXrEozCmA3SwYMHce/ePWzZsgVFRUXYtWsXKisrERAQoJp68r+lpqaivLxc7cfKykr1+MOHDyEWi3HgwAGcO3cOn3zyCR48eACJRIKysjK1tm7cuIHc3FzY29uP+a9qnz59musSjMKYvbIhMzMTTk5Oasvmzp2LadOmISsrCykpKWqPiUSiQWf9k0qlkEqlassiIyPh5OSEo0ePIjg4WLVcIpGgtbUVAHDs2DEUFxeP9HDIEPJ2hWKqz9uYHfUnnZaPNoN8R+rr60NqaiqEQiGsra0hFouhUCggEomwevVqANAIEfBiljyBQIDm5mZW6rC1tYWVlRXGjRunttzc3CCfNsIhg/yLWLlyJXbu3In4+HgUFRUhJiYGS5cuxa1btxAQEDDgdkqlEm1tbfDx8dHapoWFBRwcHBAdHY2GhgatbfT29qKnpwd3795VXT6zZs0adg7MAA00MzwZHoPr2uXk5OD48eMoLS1VnX+EhYXhypUrKCgoGDBIPT09iI+Ph0AgUL1rAcDEiRORlJSE0NBQ2NnZoa6uDnK5HEFBQaiuroanp6daOyEhIbh8+TIAYNKkSSgqKoKvr+8rHUtsbCzOnz+P7u5uiEQi7N+/H0FBQa/Ulr5cu3YNzs7OXJcx5hlckPbs2YOIiAiNk3gvLy9YWlpq/aNmGAarVq1CVVUVCgsL4ejoqHrMz88Pfn5+qt8lEgnCw8MhFoshl8uRnZ2t1lZ2djba29vR0tKCw4cPIyIiAmfOnHmlSYtlMhmOHj2KcePGobCwEEuWLMG9e/d02tbMzGzY+3vZpk2bhlwnPT19yPXS09NHXMvLlsi+xuvTQ4e1TdWXu/HN2VS1ZT3dTzHV522d21AoSrEhPEzn9XWdY8KggtTc3AylUqn1P7apqQne3t5qI239EhMTceLECeTk5CA8PHzI/QiFQgQGBqKiokLjMZFIpPr3okWLEBQUhI0bN6K2tnaYR/OfDzsZhoGlpSV++OEHdHd3w9raethtEWD2IpnWwQZDYHBBAoDJkyerLe/q6oJCocA777yjsU1SUhKysrKQnZ2tMeo2GIZhhnzVNzc3x6xZs3DkyBGd231ZbGws8vPz8ezZM6xbt07nELEx244u07qkp6erdYW1SUtLG3EtL8so4eb7SCEhocjbxf5MRgY12CAQCAAAjY2Nasv37t2L+/fvw9/fX225TCZDWloaMjIysGLFCp33c/36dVRWVg46HA68OO+6dOkSvLy8dG77ZZ999hk6OjrwxRdfDLk/LuzYsYPrEoyCQb0jeXh4YMaMGZDL5eDz+XB1dUVeXh7Onj0LAGoDDfv27YNcLodUKkVAQIBaN83Ozk7VrYqNjYWHhwf8/f1hb2+Puro6JCcng8fjQSaTqbaJiorCzJkzIRaLwefz0dTUhEOHDkGpVKKgoECj1v6779TU1AAASkpKUF9fj/HjxyMyMlJtXUtLSyxatAhisRizZ8/GG2+8wdIzNnJczNZnjAxuxr7GxkbEx8ejqqoKjo6OiIuLw4QJEyCTyfDkyRPY2NgAAEJDQwccug0JCVFNe5+cnIyTJ0/izp076OzshLOzM+bPn4/t27fD3d1dtU1KSgry8vJw8+ZNdHR0gM/nIygoCJs3b8acOXM09jFQt9DNzU3rlRXAi9nx9u/fr9N5HBsMecY+rrp2ns7A+vnst2twQdJm2bJlqK2txdWrV7kuRWcPHz7EV199hV//+tewsLDAX/7yF/zxj3/EzZs3VV1YfaMgadJXkAzqHGkgNTU1g34Qa6gyMjLg4uKCSZMm4eTJkygsLBy1EJHRZVDnSNo8ffoUjY2NWLt2LdelDIujoyP++c9/cl3GkF7l8zGiyeCDZGtri97eXq7LMFpZWVlcl2AUxkTXjuhPQkIC1yUYBQqSiesf3SQjQ0EihAUUJEJYQEEycUN9hkR0Q0EycadOneK6BKNg8MPfRL+2bdvGyfV2rg6jvku97peCRDihjzmKuERdO0JYQEEycZmZmVyXYBQoSCbO29ub6xKMAgXJxI31O8UaCgoSISygIBHCAhr+NmK6fLN127ZtevkGrKkZE181J8TQUdeOEBZQkAh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\n" }, - "execution_count": 439, "metadata": {}, - "output_type": "execute_result" + "execution_count": 9 } ], "source": [ @@ -1085,20 +643,20 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 10, "metadata": {}, "outputs": [ { "output_type": "execute_result", "data": { "text/plain": [ - "
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\n" }, "metadata": {}, - "execution_count": 24 + "execution_count": 10 } ], "source": [ @@ -1121,7 +679,7 @@ }, { "cell_type": "code", - "execution_count": 216, + "execution_count": 11, "metadata": {}, "outputs": [ { From de71b852c05c1c6e1058786d2d1183860ccc0574 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Mon, 26 Apr 2021 19:19:37 +0200 Subject: [PATCH 19/30] fix: changes --- "Mermin\342\200\223Peres-Game.ipynb" | 584 +++++++++++++++++++-------- Mirmin-Peres-Game/MPG.png | Bin 4833 -> 42303 bytes 2 files changed, 409 insertions(+), 175 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 7ade691..7f9926c 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -4,65 +4,71 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "# Mermin Peres Magic Square Game" + "# Quantum pseudo-telepathy: Mermin Peres Magic Square Game" ] }, { "source": [ - "The Mermin Peres Magic Square Game is a Game that is not solvable with a probability of 100% on classical computers. It operates in the following manner: Two players, Alice and Bob, play a cooperative game, yet unable to communicat by playing in a respectivly randomly assigned column / row either +1 or -1. The game is won if the number of +1 played in the selcted column is even and the -1 played in the rows is odd, further and crucially, the intersecting cell between column and row must be the same. As illustrated in the image below there is no classical solution to win the game with probability of 100%, however, the game can be won with a probability of 88,89%" + "Communication is bilateral. Without proper forms of communication many situations but also games cannot be resolved in the classical sense. TThe Mermin-Peres magic square game is an example of this. It was first proposed by Aravind in 2002, based on the work of Mermin, 1990 and Peres, 1990. It operates, as illustrated below, according to the following scheme: two players, Alice and Bob, play against a game master on a 3x3 grid. They not allowed to communicate. The game master chooses a row and assigns it to Bob and a column is assigned to Alice. \n", + "\n", + "The rules state that the number of tiles placed by Alice in the column must be either zero or two tiles and those placed by Bob in the row must be also either zero or two tiles. However, there must be unity with respect to the intersecting cell, between the row and the column and at least one tile of either player must be placed there. As we can see below, there are various cases when one of the two players places a tile incohrently (i.e. both play a tile in the intersecting cell) and the game is subsequeently lost, so there is no guarantee that the game can be won with a probability of 1. Formally, in a 3x3 grip there are 9 distinct cells, however as presented above the number of total tiles placed is even, as a result Alice and Bob can't agree on a non-overlapping strategy.\n", + "\n", + "However, as an equilibrium strategy Alice and Bob can construct a 3x3 table, containing static examples of moves that can be played by them so that both of them win. However, a conflict between the winning conditions of Alice and Bob is bound to emerge, such that $\\frac 8 {9}$ tiles can be placed without conflict. Consequently, the game has a winning probability of $\\frac 8 {9}$.\n" ], "cell_type": "markdown", "metadata": {} }, { "source": [ - "\n" + "

Source: [Twisted Oaks](http://twistedoakstudios.com/blog/Post6536_implementing-quantum-pseudo-telepathy)" ], "cell_type": "markdown", "metadata": {} }, { "source": [ - "However, in order to solve this game with a probility of 100%, one might look in the direction of quantum mechanics, quantum computing and quantum pseudo-telepathy where superposition and entanglement have already solved problems unfeasible on classical systems. Then, to realise this feat the Python Qiskit framework was employed, [extended from this Medium atricle] (https://medium.com/qiskit/this-proof-demonstrates-a-quantum-advantage-even-for-noisy-quantum-computers-b44a738801ad). Subsequently, this notebook will introduce the quantum circuit solving the MPG game, provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game and quantify it's performance on both simulated and real quantum computers.\n", - "\n", + "However, we can describe the game using a quantum mechanical system that gurantees to solve the game with a probability of 1. We methods such that it seems that Alice and Bob are communucating, hence the name Quantum pseudo-telepathy. This game has been [recently implemented] (https://medium.com/qiskit/this-proof-demonstrates-a-quantum-advantage-even-for-noisy-quantum-computers-b44a738801ad) and our aim is to explain it in detail and extend it to real quantum computers. In the orginal publication, rather than graphical tiles denoted with A & B as in the illustration above, conceptionally equivilant digits +1 and -1 are amployed for comptation. Hereafter, the tile B will be replaced by +1 and tile A by -1. \n", "\n", - "First, for the subsequent circuit definition, evaluation and illustration, we first have to define auxiliary functions and import the corresponding modules from Qiskit and Numpy" + "To the end of implemenation, the Python Qiskit framework was employed. Subsequently, this notebook will introduce the quantum circuit solving the MPG game, provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game and quantify it's performance on both simulated and real quantum computers. First, for the subsequent circuit definition, evaluation and illustration, we first have to define auxiliary functions and import the corresponding modules from Qiskit and Numpy" ], "cell_type": "markdown", "metadata": {} }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 35, "metadata": {}, "outputs": [], "source": [ - "from qiskit import *\n", "import numpy as np\n", - "from qiskit.test.mock import *\n", + "import random\n", + "import qiskit\n", + "\n", + "from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister, execute, BasicAer, IBMQ\n", + "from qiskit.providers.aer import QasmSimulator, UnitarySimulator\n", "from qiskit.visualization import plot_state_qsphere\n" ] }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 82, "metadata": {}, "outputs": [], "source": [ "#checks if the circuit output fulfills the winning criteria\n", "def check_result(results, col, row):\n", " \n", - " print_tut = True\n", - " expected_states = []\n", + " print_tut = True # variable governing printing of tutorial \n", + " expected_states = [] # array holding the answers used for accuracy \n", " \n", " for i in results: \n", " \n", - " #if print_tut:\n", - " # explain(i, col, row)\n", - " # print_tut = False\n", + " if print_tut: #prints explaination of transformation of the quantum circuit solution, for only the very first solution concept\n", + " explain(i, col, row)\n", + " print_tut = False\n", "\n", - " #print(\"Quantum circuit solution\")\n", - " #print(i+\"\\n\")\n", + " print(\"Quantum circuit solution\")\n", + " print(i+\"\\n\")\n", " \n", " i_orginal = i\n", " i = np.array(list(i.replace(\" \", \"\")), dtype=np.int32)[::-1]\n", @@ -83,28 +89,29 @@ " # compute y3 by y1+y2 mod 2 and save to array\n", "\n", " # compute overall winning condition -> alice at index of bob's row and vice versa\n", - " #print(\"Alice's selection. Column:\"+str(col)+\"\\n\"+str(a.reshape(3,1))+\"\\n\")\n", - " #print(\"Bob's circuit solution. Row:\"+str(row)+\"\\n\"+str(b)+\"\\n\")\n", + " print(\"Alice's selection. Column:\"+str(col)+\"\\n\"+str(a.reshape(3,1))+\"\\n\")\n", + " print(\"Bob's circuit solution. Row:\"+str(row)+\"\\n\"+str(b)+\"\\n\")\n", " \n", " if(winning_bob & winning_alice & (a[row-1]==b[col-1])):\n", - " #print(\"Victory!\") \n", - " #print(\"The following magic square wins the game!\")\n", + " print(\"Victory!\") \n", + " print(\"The following magic square wins the game!\")\n", + " print(\"\\n\")\n", "\n", " expected_states.append(str(i_orginal))\n", " k = np.empty((3,3,))\n", " k[:] = np.nan\n", " k[:,col-1] = a\n", " k[row-1,:] = b\n", - " #print(k)\n", - "\n", - " #print(\"------------------------------------------\") \n", - " #print(\"\\n\")\n", + " print(k)\n", + " print(\"\\n\")\n", + " print(\"------------------------------------------\") \n", + " print(\"\\n\")\n", " return expected_states" ] }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 113, "metadata": {}, "outputs": [], "source": [ @@ -112,12 +119,13 @@ "def explain(result, col, row):\n", " print(\"Resultant quantum circuit solution:\"+\"\\n\")\n", " print(result+\"\\n\")\n", - " example = np.array(list(result.split())).astype(int)[::-1]\n", + " example = np.array(list(result), dtype=np.int32)[::-1]\n", " print(\"Bring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\"+\"\\n\")\n", " print(str(example)+\"\\n\")\n", " print(\"Compute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\")\n", " print(\"Compute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\"+\"\\n\")\n", " \n", + "\n", " a = example[0:2]\n", " a_sum = np.sum(a)\n", " a3 = int(np.mod(a_sum+1,2))\n", @@ -130,6 +138,7 @@ " b = np.append(b,b3)\n", " winning_bob = np.mod(b_sum + b3, 2) == 0 \n", "\n", + " print(a.shape)\n", " print(\"Alice's column solution\"+\"\\n\"+str(a.reshape(3,1))+\"\\n\" + \"Winning conditions fullfilled (Odd quantitude of 1)?: \"+str(winning_alice)+\"\\n\")\n", " print(\"Bob's row solution\"+\"\\n\"+str(b)+\"\\n\"+ \"Winning conditions fullfilled (Even quantitude of 1)?: \"+str(winning_bob)+\"\\n\")\n", " print(\"Create matrix to show results\"+\"\\n\")\n", @@ -138,41 +147,29 @@ " k[:,col-1] = a\n", " k[row-1,:] = b\n", " print(k)\n", + " print(\"\\n\")\n", " print(\"------------------------------------------\") \n", - " print(\"------------------------------------------\") " - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "metadata": {}, - "outputs": [], - "source": [ - "# function to calculate state accuracy\n", - "def get_state_accuracy(counts, expected_states, shots): \n", - " expected_counts = 0\n", - " for state in counts.keys():\n", - " if state in expected_states:\n", - " expected_counts = expected_counts + counts[state]\n", - " state_accuracy = expected_counts / shots\n", - " return str(state_accuracy*100)+\"%\"" + " print(\"------------------------------------------\") \n", + " print(\"\\n\")" ] }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 114, "metadata": {}, "outputs": [], "source": [ "# function that sets up the initial entanglement between Alice and Bob\n", - "def share_bell_state(qc,a,b,c,d): \n", - " qc.h(a)\n", - " qc.h(b)\n", - " qc.cx(a,c)\n", - " qc.cx(b,d)\n", + "def share_bell_state(qc): \n", + " qc.h(0)\n", + " qc.h(1)\n", + " qc.cx(0,2)\n", + " qc.cx(1,3)\n", " \n", "# The following functions represent the U(gamma) and V(gamma) controlled Cliffords.\n", - "def U(qc,gamma,a,b):\n", + "def U(qc,gamma):\n", + " a = 0\n", + " b = 1\n", " if gamma==1:\n", " qc.h(a)\n", " qc.i(b)\n", @@ -187,7 +184,9 @@ " qc.h(a)\n", " qc.i(b)\n", "\n", - "def V(qc,gamma,a,b):\n", + "def V(qc,gamma):\n", + " a = 2\n", + " b = 3\n", " if gamma==1:\n", " qc.h(a)\n", " qc.h(b)\n", @@ -205,79 +204,282 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 117, "metadata": {}, "outputs": [ { "output_type": "stream", "name": "stdout", "text": [ - "Alice and Bob, select a column or row value of 1, 2, or 3\n", - "Alice choice (column):\n", - "Bob choice (row):\n", - " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", - "q0_0: ┤ H ├──■────────░───■──┤ H ├──────░─┤M├─────────\n", - " ├───┤ │ ░ ┌─┴─┐├───┤ ░ └╥┘┌─┐ \n", - "q0_1: ┤ H ├──┼────■───░─┤ X ├┤ I ├──────░──╫─┤M├──────\n", - " └───┘┌─┴─┐ │ ░ ├───┤└───┘┌───┐ ░ ║ └╥┘┌─┐ \n", - "q1_0: ─────┤ X ├──┼───░─┤ Z ├──■──┤ H ├─░──╫──╫─┤M├───\n", - " └───┘┌─┴─┐ ░ ├───┤ │ ├───┤ ░ ║ ║ └╥┘┌─┐\n", - "q1_1: ──────────┤ X ├─░─┤ Z ├──■──┤ H ├─░──╫──╫──╫─┤M├\n", - " └───┘ ░ └───┘ └───┘ ░ ║ ║ ║ └╥┘\n", - "c0: 1/═════════════════════════════════════╩══╬══╬══╬═\n", - " 0 ║ ║ ║ \n", - " ║ ║ ║ \n", - "c1: 1/════════════════════════════════════════╩══╬══╬═\n", - " 0 ║ ║ \n", - " ║ ║ \n", - "c2: 1/═══════════════════════════════════════════╩══╬═\n", - " 0 ║ \n", - " ║ \n", - "c3: 1/══════════════════════════════════════════════╩═\n", - " 0 \n", + "Alice and Bob, are assigned a column or row value of 1, 2, or 3\n", + "Alice assignment (column):\n", + "Bob assignment (row):\n", + "3\n", + " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", + "q899_0: ┤ H ├──■────────░───■──┤ H ├──────░─┤M├─────────\n", + " ├───┤ │ ░ ┌─┴─┐├───┤ ░ └╥┘┌─┐ \n", + "q899_1: ┤ H ├──┼────■───░─┤ X ├┤ I ├──────░──╫─┤M├──────\n", + " └───┘┌─┴─┐ │ ░ ├───┤└───┘┌───┐ ░ ║ └╥┘┌─┐ \n", + "q899_2: ─────┤ X ├──┼───░─┤ Z ├──■──┤ H ├─░──╫──╫─┤M├───\n", + " └───┘┌─┴─┐ ░ ├───┤ │ ├───┤ ░ ║ ║ └╥┘┌─┐\n", + "q899_3: ──────────┤ X ├─░─┤ Z ├──■──┤ H ├─░──╫──╫──╫─┤M├\n", + " └───┘ ░ └───┘ └───┘ ░ ║ ║ ║ └╥┘\n", + " c76: 4/═════════════════════════════════════╩══╩══╩══╩═\n", + " 0 1 2 3 \n", + "{'0001': 1024, '1110': 971, '1011': 1000, '0100': 1041, '1101': 1063, '0111': 1003, '1000': 1054, '0010': 1036}\n", + "Resultant quantum circuit solution:\n", + "\n", + "0001\n", + "\n", + "Bring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", + "\n", + "[1 0 0 0]\n", + "\n", + "Compute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\n", + "Compute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n", + "\n", + "(3,)\n", + "Alice's column solution\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "Winning conditions fullfilled (Odd quantitude of 1)?: True\n", + "\n", + "Bob's row solution\n", + "[0 0 0]\n", + "Winning conditions fullfilled (Even quantitude of 1)?: True\n", + "\n", + "Create matrix to show results\n", + "\n", + "[[nan nan 1.]\n", + " [nan nan 0.]\n", + " [ 0. 0. 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0001\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[0 0 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan nan 1.]\n", + " [nan nan 0.]\n", + " [ 0. 0. 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1110\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[1 1 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan nan 0.]\n", + " [nan nan 1.]\n", + " [ 1. 1. 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1011\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[0 1 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan nan 1.]\n", + " [nan nan 1.]\n", + " [ 0. 1. 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0100\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[1 0 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan nan 0.]\n", + " [nan nan 0.]\n", + " [ 1. 0. 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1101\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[1 1 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan nan 1.]\n", + " [nan nan 0.]\n", + " [ 1. 1. 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0111\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[1 0 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan nan 1.]\n", + " [nan nan 1.]\n", + " [ 1. 0. 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1000\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[0 1 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan nan 0.]\n", + " [nan nan 0.]\n", + " [ 0. 1. 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0010\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[0 0 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan nan 0.]\n", + " [nan nan 1.]\n", + " [ 0. 0. 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", "100.0%\n" ] } ], "source": [ "#We will ask the user to select the row and column on Alice and Bob's behalf.\n", - "print(\"Alice and Bob, select a column or row value of 1, 2, or 3\") \n", - "print(\"Alice choice (column):\")\n", - "alpha = int(input())\n", - "print(\"Bob choice (row):\")\n", - "beta = int(input())\n", - "\n", + "print(\"Alice and Bob, are assigned a column or row value of 1, 2, or 3\") \n", + "print(\"Alice assignment (column):\")\n", + "alpha = random.randint(1, 4)\n", + "print(\"Bob assignment (row):\")\n", + "beta = random.randint(1, 4)\n", + "print(alpha)\n", "#Create the quantum register and the classical register to store our final bit values\n", - "aliceQR = QuantumRegister(2)\n", - "x1CR = ClassicalRegister(1)\n", - "x2CR = ClassicalRegister(1)\n", - "bobQR = QuantumRegister(2)\n", - "y1CR = ClassicalRegister(1)\n", - "y2CR = ClassicalRegister(1)\n", + "q = QuantumRegister(4) # Alice-Qunatum Register (2) and Bob-Qunatum Register (2)\n", + "c = ClassicalRegister(4) \n", "\n", "#Create the circuit\n", - "magicsquare_circuit = QuantumCircuit(aliceQR,bobQR,x1CR,x2CR,y1CR,y2CR)\n", + "magicsquare_circuit = QuantumCircuit(q,c)\n", "\n", "#Generate the Bell state on the circuit\n", - "share_bell_state(magicsquare_circuit,0,1,2,3)\n", - "\n", + "share_bell_state(magicsquare_circuit)\n", "magicsquare_circuit.barrier()\n", "\n", "#Draw the rest of the circuit based on Alice and Bob's selection \n", "if 4>alpha>0 and 4>beta>0:\n", - " U(magicsquare_circuit,alpha,0,1) \n", - " V(magicsquare_circuit,beta,2,3)\n", + " U(magicsquare_circuit,alpha) \n", + " V(magicsquare_circuit,beta)\n", " magicsquare_circuit.barrier()\n", - " magicsquare_circuit.measure(0,0)\n", - " magicsquare_circuit.measure(1,1)\n", - " magicsquare_circuit.measure(2,2)\n", - " magicsquare_circuit.measure(3,3)\n", + " magicsquare_circuit.measure(q,c)\n", " \n", " magicsquare_circuit.draw(output='mpl')\n", " print(magicsquare_circuit) \n", "\n", " job = execute(magicsquare_circuit, BasicAer.get_backend('qasm_simulator'), optimization_level=3, shots=8192) # run the job simulation\n", " results = job.result().get_counts()\n", + " print(results)\n", " expected_states = check_result(results, alpha, beta)\n", " state_accuracy = get_state_accuracy(results,expected_states,8192)\n", " print(state_accuracy)\n", @@ -286,6 +488,22 @@ " print(\"please enter values of 1, 2, or 3\")" ] }, + { + "cell_type": "code", + "execution_count": 88, + "metadata": {}, + "outputs": [], + "source": [ + "# function to calculate state accuracy\n", + "def get_state_accuracy(counts, expected_states, shots): \n", + " expected_counts = 0\n", + " for state in counts.keys():\n", + " if state in expected_states:\n", + " expected_counts = expected_counts + counts[state]\n", + " state_accuracy = expected_counts / shots\n", + " return str(state_accuracy*100)+\"%\"" + ] + }, { "source": [ "#### Indeed, using an ideal quantum simulator, a winning probability of 100% is achieved, which exceeds the highest classical winning probability of 88.99% Beyond that, however, it is desirable to quantify the results on a real IBM quantum computer, which will be realized in the last section." @@ -299,17 +517,15 @@ "source": [ "# Mathmatical background of the computation\n", "\n", - "As we have demonstrated above, the quantum circuit solves the Mirmin-Peres game (MPG) on real quantum computers with a higher probability than with any classical strategy on ordinary computers. To shed some light on how and why this works, we present and explain the mathematical operations that lead to the solution of this game. In particular, we will first present a high-level equation describing the mathematical rationale and then we will take a look at each element of this equation explain it step by step.\n", - "\n", + "As we have demonstrated above, the quantum circuit solves the Mirmin-Peres game (MPG) on simulators with a higher probability than with any classical strategy on ordinary computers possible. To shed some light on how and why this works, we present and explain the mathematical operations that lead to the solution of this game. In particular, we will first present a high-level equation describing the mathematical rationale and then we will take a look at each element of this equation explain it step by step.\n", "\n", - "Recall that the quantum circuit has to play different moves for each different row/column instances of the MPG being selected, since there is no classical strategy that can win the game with probability = 1. Thus, to win with probability 1, the quantum circuit dynamically adapts to the input of each column and row played. Nevertheless, there is indeed one part of the circuit that remains static and is responsible for generating a matching qubit state and we will start of by focusing on that part. \n", "\n", - "The following Pauli operators govern what the ideal mathmatical operations, with the winning criteria in mind, are in correspondace to the selected row and column. \n", + "Recall that the quantum circuit has to play different moves for each different row/column instances of the MPG being selected, since there is no classical strategy that can win the game with probability = 1. Thus, to win with probability 1, the quantum circuit dynamically adapts to the input of each column and row played. Nevertheless, there is indeed one part of the circuit that remains static and is responsible for generating a matching qubit state, that entangles the qubits of Alice and Bob. As this part comes prior to any adaptation we will start off by focusing on that part. \n", "\n", "![title](Mirmin-Peres-Game/Pauli_Operators.png)\n", "\n", "\n", - "These operators can then be translated into the following quantum gates for the respective row / column selected. To further understand why these Pauli operators are translated to the gates, see.: [Quantum advantage in noisy flat circuits](https://arxiv.org/pdf/1904.01502.pdf) \n", + "These operators can then be translated into the [following quantum gates] (https://arxiv.org/pdf/1904.01502.pdf) for the r(spective row / column selected. To further understand why these Pauli operators are translated to the gates, see.: [Quantum advantage in noisy flat circuits](https://arxiv.org/pdf/1904.01502.pdf) \n", "\n", "![title](Mirmin-Peres-Game/Circuit_Operations.png)\n", "\n", @@ -325,45 +541,54 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 38, "metadata": {}, "outputs": [ { "output_type": "execute_result", "data": { "text/plain": [ - "
" + "
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}, "metadata": {}, - "execution_count": 8 + "execution_count": 38 } ], "source": [ "q = QuantumRegister(4) \n", "c = ClassicalRegister(4) \n", "qc = QuantumCircuit(q, c) \n", - "qc.h(q[0])\n", - "qc.h(q[1])\n", - "qc.cx(q[0],q[2])\n", - "qc.cx(q[1],q[3])\n", - "#qc.barrier()\n", + "share_bell_state(qc,0,1,2,3)\n", "qc.draw(output='mpl')\n" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### This initial vector is generated, as illustrated above, by the application of two Hadamard gates to the first and second qubits and CNOT gates, with the first and second qubits serving as the control qubit and the third and fourth qubits, respectively, serving as the target qubit. Mathematically, these operations can be described by the following equation: \n", + "\n", + "$$ Initial Vector = (I ⨂ CNOT_{0|2}) ⨂ (CNOT_{1|3} ⨂ I) * (I ⨂ I ⨂ H ⨂ H) * InitialQubits $$\n", + "\n", + "
\n", + "\n", + "#### However, prior to inserting the corresponding matrices into this equation, we introduce the relevant matrices needed for the computation. " + ] + }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### The very starting vector state of the MPG circuit, prior to any quantum gate interaction, corresponds to the Dirac bracket |0000> and is hence described by the following vector: \n", - "$$ InitialQubits= \\frac 1 {\\sqrt{2}} {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}$$" + "$$ InitialQubits= {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}$$" ] }, { "source": [ - "#### In terms of the gates, these are the mathmatical represetations of the relevant gates illustrated above:\n", + "#### In terms of the gates, these are the mathmatical represetations of the relevant gates illustrated above. \n", "\n", "$$\n", "I = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} \n", @@ -383,21 +608,22 @@ "0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ \n", "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\ \n", "\\end{pmatrix}\n", - "$$" + "$$\n", + "
\n", + "The \"i\" in CNOT(i, i+2) refers to the control qubit in the first instance and to the target qubit in the second instance under the condition that there are at least two qubits in between such as $CNOT_(1,3)$\n", + "\n" ], "cell_type": "markdown", "metadata": {} }, { - "cell_type": "markdown", - "metadata": {}, "source": [ - "#### While the static matrix, represented as the quantum gates in the illustration above, is created by the appliance of two CNOT gates to the first and third qubit and the second and fourth qubit, as well as two Hadamard gates applied to the first and second qubit. Mathematically, this is described by the following equation: \n", - "\n", - "$$ Initial Vector = (I ⨂ CNOT_{0|2}) ⨂ (CNOT_{1|3} ⨂ I) * (I ⨂ I ⨂ H ⨂ H) * InitialQubits $$\n", + "#### Now we can recall the equation of the initial vectors and insert the matrices:\n", "\n", - "Subsequently inserting the corresponding matrices into this equation results in the following matrix:" - ] + "$$ Initial Vector = (I ⨂ CNOT_{0|2}) ⨂ (CNOT_{1|3} ⨂ I) * (I ⨂ I ⨂ H ⨂ H) * InitialQubits $$\n" + ], + "cell_type": "markdown", + "metadata": {} }, { "cell_type": "markdown", @@ -407,7 +633,7 @@ "Inital Vector = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} *\n", "\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} \n", "*\n", - "\\frac 1 {\\sqrt{2}} {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", + "{\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", "$$\n", "\n", "$$ \n", @@ -419,7 +645,7 @@ "\n", "\\frac 1 {2} \\begin{pmatrix} 1 & 1 & 1 & 1\\\\ 1 & -1 & 1 & -1 \\\\ 1 & 1 & -1 & -1\\\\ 1 & -1 & -1 & 1 \\end{pmatrix}\n", "*\n", - "\\frac 1 {\\sqrt{2}} {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", + "{\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", "$$" ] }, @@ -432,13 +658,13 @@ "*\n", "\\begin{bmatrix}\\ \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} \\\\\\end{bmatrix}\\\n", "*\n", - "\\frac 1 {\\sqrt{2}} {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", + "{\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", "$$\n", "\n", "$$\n", "=\\begin{bmatrix}\\ \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2}& 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} \\\\\\ \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} & \\frac 1 {2} \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & \\frac 1 {2} & \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ \\frac 1 {2} & -\\frac 1 {2} & -\\frac 1 {2} & \\frac 1 {2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", "*\n", - "\\frac 1 {\\sqrt{2}} {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", + "{\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", "$$\n", "$$\n", "= \\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\ 0\\\\ \\frac 1 {2}\\end{pmatrix}\n", @@ -449,7 +675,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "#### This vector holds the relevant qubits that can be further considered and modified in the dynmaic part of the circuit so that valid winning can be created, these Qubit states are:\n", + "#### This resulting vector holds the relevant qubit states that can be further considered and modified in the dynmaic part of the circuit so that valid winning condition can be created, these Qubit states are:\n", "$$= \\frac 1 {2} (|0000\\rangle+|0101\\rangle+|1010\\rangle+|1111\\rangle)$$\n" ] }, @@ -460,12 +686,12 @@ "#### Furthermore, when analyzing the initial vector, it becomes apparent that when multiplying any matrix by it, only four relevenate cells per row in the multiplier remain open for consideration, since the others are all multiplied by 0 and thereby likewise drop of with 0% probability as a result. Thus, we can eliminate the majority of the entries whatsoever. As an example we have prepared the matrix below, and denoted all entries with an i, that have to be computed and all others that result in any case with 0 with -.\n", "\n", "$$ = \n", - "\\begin{bmatrix}\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\ \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} & - & - & - & - & \\color{red}{i} \\\\\\end{bmatrix} \n", + "\\begin{bmatrix}\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\end{bmatrix} \n", "*\n", "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\ \\frac 1 {2}\\end{pmatrix}\n", "$$\n", "\n", - "Let us now take a look at an concrete example, which show and proof that only the i-rows are relevant for the subsequent computation:\n", + "Let us now take a look at an concrete example, which shows and proofs that only the Xth-columns are relevant for the subsequent computation. Since all other columns are multiplied by zero and thus remain zero:\n", "\n", "$$\n", "Winnning Plays= \n", @@ -501,35 +727,29 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 39, "metadata": {}, "outputs": [ { "output_type": "execute_result", "data": { "text/plain": [ - "
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\n" }, "metadata": {}, - "execution_count": 9 + "execution_count": 39 } ], "source": [ "q = QuantumRegister(4) \n", "c = ClassicalRegister(4) \n", "qc = QuantumCircuit(q, c) \n", - "#qc.h(q[0])\n", - "#qc.h(q[1])\n", - "#qc.cx(q[1],q[3])\n", - "#qc.cx(q[0],q[2])\n", "qc.barrier()\n", - "qc.h(q[0])\n", - "qc.id(q[1])\n", - "qc.h(q[2])\n", - "qc.h(q[3])\n", + "U(qc,1,0,1) \n", + "V(qc,1,2,3)\n", "qc.draw(output='mpl')" ] }, @@ -636,50 +856,44 @@ "source": [ "#### Finally, this vector contains the qubits that can be played to win the game with probability 1, these are:\n", "\n", - "$$= \\frac 1 {8} (|0000\\rangle+|0100\\rangle+|0101\\rangle+|01111\\rangle+|1000\\rangle)+|1011\\rangle)+|1101\\rangle)+|1111\\rangle)$$\n", + "$$= \\frac 1 {\\sqrt{8}} (|0000\\rangle+|0100\\rangle+|0101\\rangle+|01111\\rangle+|1000\\rangle)-|1011\\rangle)+|1101\\rangle)-|1111\\rangle)$$\n", "\n", "#### To validate these mathematical results, we will run the quantum circuit in one go and compare the respective qubit results" ] }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 40, "metadata": {}, "outputs": [ { "output_type": "execute_result", "data": { "text/plain": [ - "
" + "
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\n" }, "metadata": {}, - "execution_count": 10 + "execution_count": 40 } ], "source": [ "q = QuantumRegister(4) \n", "c = ClassicalRegister(4) \n", "qc = QuantumCircuit(q, c) \n", - "qc.h(q[0])\n", - "qc.h(q[1])\n", - "qc.cx(q[1],q[3])\n", - "qc.cx(q[0],q[2])\n", + "share_bell_state(qc,0,1,2,3)\n", "qc.barrier()\n", "U(qc,1,0,1) \n", "V(qc,1,2,3)\n", - "qc.measure(0,0)\n", - "qc.measure(1,1)\n", - "qc.measure(2,2)\n", - "qc.measure(3,3)\n", + "qc.measure(q,c)\n", "qc.draw(output='mpl')" ] }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 42, "metadata": {}, "outputs": [ { @@ -691,7 +905,7 @@ } ], "source": [ - "backend = Aer.get_backend('qasm_simulator')\n", + "backend = BasicAer.get_backend('qasm_simulator')\n", "results = execute(qc, backend, optimization_level=3, shots=8192).result().get_counts()\n", "expected_states = check_result(results, 1, 1)\n", "state_accuracy = get_state_accuracy(results,expected_states,8192)\n", @@ -710,12 +924,17 @@ "------------------------------------------------------------------------------------------------------------------------------------------------------------------\n", "To reproduce these operations for other configurations, it is recommended to use the Unitary simulator - follow this simple guide:\n", "\n", - "1. Install \"pip install git+https://github.com/qiskit-community/qiskit-textbook.git#subdirectory=qiskit-textbook-src\"\n", + "1. Install additional libararies:
\n", + "\"!pip install \"git+https://github.com/qiskit-community/qiskit-textbook.git#subdirectory=qiskit-textbook-src\"\"\n", "\n", - "2. Create new Cell & Fill:\n", + "2. Imports libararies:
\n", "from qiskit_textbook.tools import array_to_latex;\n", - "result = execute(qc, Aer.get_backend('unitary_simulator')).result().get_unitary(qc);\n", - "array_to_latex(result, pretext=\"\\\\text{Circuit = }\\n\");\n", + "from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister, BasicAer;\n", + "\n", + "3. Intialise a n-Qubit system and apply gates, an example 2-Qubit system:
\n", + "qc = QuantumCircuit(QuantumRegister(2) , ClassicalRegister(2))
\n", + "result = execute(qc, BasicAer.get_backend('unitary_simulator')).result().get_unitary(qc)
\n", + "array_to_latex(result, pretext=\"\\\\text{Circuit = }\\n\") \n", "------------------------------------------------------------------------------------------------------------------------------------------------------------------" ], "cell_type": "markdown", @@ -739,14 +958,14 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 43, "metadata": {}, "outputs": [ { "output_type": "stream", "name": "stdout", "text": [ - "95.27587890625%\n" + "92.88330078125%\n" ] } ], @@ -760,21 +979,25 @@ }, { "source": [ - "#### Now in order to evaluate the performance of real devices we select the ibmq_athens computer to benchmark the MPG. Please note that you will have to add your own API token from the IBM Quantum website (https://quantum-computing.ibm.com) so that this cell executes" + "#### We can oberserve that simulated noisy devices outperform classical systems and thus indicate that real quantum computers are capable of doing the same. Now in order to evaluate the performance of real devices we select the ibmq_athens computer to benchmark the MPG. Please note that you will have to add your own API token from the IBM Quantum website (https://quantum-computing.ibm.com) so that this cell executes" ], "cell_type": "markdown", "metadata": {} }, { "cell_type": "code", - "execution_count": 53, + "execution_count": 44, "metadata": {}, "outputs": [ { - "name": "stderr", - "output_type": "stream", - "text": [ - "configrc.store_credentials:WARNING:2021-03-15 09:28:25,642: Credentials already present. Set overwrite=True to overwrite.\n" + "output_type": "error", + "ename": "NameError", + "evalue": "name 'IBMQ' is not defined", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mIBMQ\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msave_account\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'YOUR_API_TOKEN'\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;31m# retrieve from your IBM-Account: https://quantum-computing.ibm.com\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", + "\u001b[0;31mNameError\u001b[0m: name 'IBMQ' is not defined" ] } ], @@ -784,9 +1007,21 @@ }, { "cell_type": "code", - "execution_count": 392, + "execution_count": 45, "metadata": {}, - "outputs": [], + "outputs": [ + { + "output_type": "error", + "ename": "NameError", + "evalue": "name 'IBMQ' is not defined", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;31m# load IBM Quantum Experience account\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mIBMQ\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mload_account\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 3\u001b[0m \u001b[0mprovider\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mIBMQ\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_provider\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'ibm-q'\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;31m# get provider\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mNameError\u001b[0m: name 'IBMQ' is not defined" + ] + } + ], "source": [ "# load IBM Quantum Experience account \n", "IBMQ.load_account()\n", @@ -876,9 +1111,8 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" + "name": "python385jvsc74a57bd09051d1f6526b9502b8740a4800e72d224ca6abc2bf606f16227017c303a32110", + "display_name": "Python 3.8.5 64-bit ('base': conda)" }, "language_info": { "codemirror_mode": { diff --git a/Mirmin-Peres-Game/MPG.png b/Mirmin-Peres-Game/MPG.png index 82aac298fdf993f72acc9acdea4cb14562867a09..2d708ec5c764ad017be6361ee91167b9a828a5e1 100644 GIT binary patch literal 42303 zcmb@u1yr5Q(k{4h4;y!Pf_uaLcp?{m?gYP%~8az`KroTpSSv6Vzp`Bt~nu6lc{XH{^G$r(UM!}?~`@p zBw9Nx>>a;$H!&G#^bk1t@P5L&4~zvooap@=geN$Z)YR1H1Xz^VK+#ncC;npX`>o6j6%0#qOp@{O&Scc*>Zza$q&;85@ ztwur^&&>wyEQ}f?4cYv!XgL6OSOY6Ypz3SFx~R+=^x-s6-q)|H`Hv+u$eCWlQK^ElN)1nZ+) z88sG~A(WC7dLECN5Y~06Q`H8FFua2BILf?aaCx))7)WiwbY`gqB=cEq)~g; 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z%98En-)+w|sLpg#Y_ItT@8#lx3h%{8!+ebeUDVaWFp+B4^5ZUky2n2!jtIyj;_e#n zk+`HP=wK}8ckgbPX3R9DL+sNjl>uMAUw=Ip=Ve^S9eML$qE{kna^~!d7<%fwC5YKW z|NL5+St7z?7rqh9&uxkQ=ShH`eK4PTwY!3pg#o4yX7lo0++JZ@;#z=V*|Gf!`#dMh zYH`3MVuC%NDF?wTsi?ktufCfdA9eq zC4J0HUvKx$)K}9A^EqzgS&%oil*YwP&bFyJfkR4C`Lo(%cjr5cae#2KgDc{C%ak|S zKirUZ#R5|q(>fFQQdP+Ee(ebmriw`GCigw56c{fTvHI5FFX#@dbtAhAfo~yw+IVJ6m*;j;bTB;Gm`yX9g2QTbLc!7!jN8_;%>T^g5u#g z(01zau>j?mc)_dVh#$bZwe1Va#Zr=LWQ3jj<;W)hcR)NfP+4m)I!=Z(OuWdfk_SRb zK)fJRu(x2Y1ywnUompydu703|?uZ2s-r9us6=hJ-#bnzDRks(qs=NeXYJu)2=Y<=e zxv2mgli(z|Es6!bm{o(fzVs9~9@?g_M?}5fuW0_PASONfwxIaa_r?Jh9b8yTW1yJWjV}=bgcQC2-9BGGzmZ+LwqCUQBFDKNM7M+l`rJHr&U9^dsGn#)cLQ}KGE~5_>upo3{aq*e`&mVn<|4s z^tD-B@QIWgSjzvcn(do4m4}}mku^fiwPWqPVuJ%>L#RQALr?}rWJ7a(L$bc1 x;ZCwK)x?NOHY1YBR5Cfu*EINd1_vX9!}lfpcY|3>=HP08PIj)gHP)Q8e*wFT5oQ1Y From 752e6b2bd47a969921a0c7192b2ddbd4b0e0a930 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Thu, 20 May 2021 00:52:35 +0200 Subject: [PATCH 20/30] least busy device --- "Mermin\342\200\223Peres-Game.ipynb" | 1133 ++++++++++++++++++++++---- 1 file changed, 967 insertions(+), 166 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 7f9926c..ee2cd09 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -36,22 +36,38 @@ }, { "cell_type": "code", - "execution_count": 35, + "execution_count": 38, "metadata": {}, - "outputs": [], + "outputs": [ + { + "output_type": "error", + "ename": "ImportError", + "evalue": "cannot import name 'FakeVigo' from 'qiskit' (/Users/david.drexlin@ibm.com/opt/anaconda3/lib/python3.8/site-packages/qiskit/__init__.py)", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mImportError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 3\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mqiskit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 5\u001b[0;31m \u001b[0;32mfrom\u001b[0m \u001b[0mqiskit\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mQuantumCircuit\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mClassicalRegister\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mQuantumRegister\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexecute\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mBasicAer\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mIBMQ\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mFakeVigo\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 6\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mqiskit\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mproviders\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0maer\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mQasmSimulator\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mUnitarySimulator\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 7\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mqiskit\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mvisualization\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mplot_state_qsphere\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mImportError\u001b[0m: cannot import name 'FakeVigo' from 'qiskit' (/Users/david.drexlin@ibm.com/opt/anaconda3/lib/python3.8/site-packages/qiskit/__init__.py)" + ] + } + ], "source": [ "import numpy as np\n", "import random\n", "import qiskit\n", "\n", - "from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister, execute, BasicAer, IBMQ\n", + "from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister, execute, BasicAer, IBMQ, FakeVigo\n", "from qiskit.providers.aer import QasmSimulator, UnitarySimulator\n", - "from qiskit.visualization import plot_state_qsphere\n" + "from qiskit.visualization import plot_state_qsphere\n", + "from qiskit.test.mock import FakeVigo\n", + "from qiskit.providers.ibmq import least_busy\n", + "from qiskit.ignis.mitigation.measurement import complete_meas_cal,CompleteMeasFitter \n", + "from qiskit.visualization import plot_histogram " ] }, { "cell_type": "code", - "execution_count": 82, + "execution_count": 5, "metadata": {}, "outputs": [], "source": [ @@ -111,7 +127,7 @@ }, { "cell_type": "code", - "execution_count": 113, + "execution_count": 6, "metadata": {}, "outputs": [], "source": [ @@ -155,7 +171,23 @@ }, { "cell_type": "code", - "execution_count": 114, + "execution_count": 9, + "metadata": {}, + "outputs": [], + "source": [ + "# function to calculate state accuracy\n", + "def get_state_accuracy(counts, expected_states, shots): \n", + " expected_counts = 0\n", + " for state in counts.keys():\n", + " if state in expected_states:\n", + " expected_counts = expected_counts + counts[state]\n", + " state_accuracy = expected_counts / shots\n", + " return str(state_accuracy*100)+\"%\"" + ] + }, + { + "cell_type": "code", + "execution_count": 7, "metadata": {}, "outputs": [], "source": [ @@ -204,7 +236,7 @@ }, { "cell_type": "code", - "execution_count": 117, + "execution_count": 8, "metadata": {}, "outputs": [ { @@ -215,33 +247,33 @@ "Alice assignment (column):\n", "Bob assignment (row):\n", "3\n", - " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", - "q899_0: ┤ H ├──■────────░───■──┤ H ├──────░─┤M├─────────\n", - " ├───┤ │ ░ ┌─┴─┐├───┤ ░ └╥┘┌─┐ \n", - "q899_1: ┤ H ├──┼────■───░─┤ X ├┤ I ├──────░──╫─┤M├──────\n", - " └───┘┌─┴─┐ │ ░ ├───┤└───┘┌───┐ ░ ║ └╥┘┌─┐ \n", - "q899_2: ─────┤ X ├──┼───░─┤ Z ├──■──┤ H ├─░──╫──╫─┤M├───\n", - " └───┘┌─┴─┐ ░ ├───┤ │ ├───┤ ░ ║ ║ └╥┘┌─┐\n", - "q899_3: ──────────┤ X ├─░─┤ Z ├──■──┤ H ├─░──╫──╫──╫─┤M├\n", - " └───┘ ░ └───┘ └───┘ ░ ║ ║ ║ └╥┘\n", - " c76: 4/═════════════════════════════════════╩══╩══╩══╩═\n", - " 0 1 2 3 \n", - "{'0001': 1024, '1110': 971, '1011': 1000, '0100': 1041, '1101': 1063, '0111': 1003, '1000': 1054, '0010': 1036}\n", + " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", + "q0_0: ┤ H ├──■────────░───■──┤ H ├─░─┤M├─────────\n", + " ├───┤ │ ░ ┌─┴─┐├───┤ ░ └╥┘┌─┐ \n", + "q0_1: ┤ H ├──┼────■───░─┤ X ├┤ I ├─░──╫─┤M├──────\n", + " └───┘┌─┴─┐ │ ░ ├───┤└───┘ ░ ║ └╥┘┌─┐ \n", + "q0_2: ─────┤ X ├──┼───░─┤ H ├──────░──╫──╫─┤M├───\n", + " └───┘┌─┴─┐ ░ ├───┤ ░ ║ ║ └╥┘┌─┐\n", + "q0_3: ──────────┤ X ├─░─┤ H ├──────░──╫──╫──╫─┤M├\n", + " └───┘ ░ └───┘ ░ ║ ║ ║ └╥┘\n", + "c0: 4/════════════════════════════════╩══╩══╩══╩═\n", + " 0 1 2 3 \n", + "{'0010': 1014, '1011': 1009, '0111': 983, '1100': 998, '0000': 1098, '1110': 1020, '1001': 1020, '0101': 1050}\n", "Resultant quantum circuit solution:\n", "\n", - "0001\n", + "0010\n", "\n", "Bring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", "\n", - "[1 0 0 0]\n", + "[0 1 0 0]\n", "\n", "Compute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\n", "Compute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n", "\n", "(3,)\n", "Alice's column solution\n", - "[[1]\n", - " [0]\n", + "[[0]\n", + " [1]\n", " [0]]\n", "Winning conditions fullfilled (Odd quantitude of 1)?: True\n", "\n", @@ -251,9 +283,9 @@ "\n", "Create matrix to show results\n", "\n", - "[[nan nan 1.]\n", - " [nan nan 0.]\n", - " [ 0. 0. 0.]]\n", + "[[ 0. 0. 0.]\n", + " [nan nan 1.]\n", + " [nan nan 0.]]\n", "\n", "\n", "------------------------------------------\n", @@ -261,190 +293,200 @@ "\n", "\n", "Quantum circuit solution\n", - "0001\n", + "0010\n", "\n", "Alice's selection. Column:3\n", - "[[1]\n", - " [0]\n", + "[[0]\n", + " [1]\n", " [0]]\n", "\n", - "Bob's circuit solution. Row:3\n", + "Bob's circuit solution. Row:1\n", "[0 0 0]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", "\n", "\n", - "[[nan nan 1.]\n", - " [nan nan 0.]\n", - " [ 0. 0. 0.]]\n", + "[[ 0. 0. 0.]\n", + " [nan nan 1.]\n", + " [nan nan 0.]]\n", "\n", "\n", "------------------------------------------\n", "\n", "\n", "Quantum circuit solution\n", - "1110\n", + "1011\n", "\n", "Alice's selection. Column:3\n", - "[[0]\n", + "[[1]\n", " [1]\n", - " [0]]\n", + " [1]]\n", "\n", - "Bob's circuit solution. Row:3\n", - "[1 1 0]\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", "\n", "\n", - "[[nan nan 0.]\n", + "[[ 0. 1. 1.]\n", " [nan nan 1.]\n", - " [ 1. 1. 0.]]\n", + " [nan nan 1.]]\n", "\n", "\n", "------------------------------------------\n", "\n", "\n", "Quantum circuit solution\n", - "1011\n", + "0111\n", "\n", "Alice's selection. Column:3\n", "[[1]\n", " [1]\n", " [1]]\n", "\n", - "Bob's circuit solution. Row:3\n", - "[0 1 1]\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", "\n", "\n", - "[[nan nan 1.]\n", + "[[ 1. 0. 1.]\n", " [nan nan 1.]\n", - " [ 0. 1. 1.]]\n", + " [nan nan 1.]]\n", "\n", "\n", "------------------------------------------\n", "\n", "\n", "Quantum circuit solution\n", - "0100\n", + "1100\n", "\n", "Alice's selection. Column:3\n", "[[0]\n", " [0]\n", " [1]]\n", "\n", - "Bob's circuit solution. Row:3\n", - "[1 0 1]\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", "\n", "\n", - "[[nan nan 0.]\n", + "[[ 1. 1. 0.]\n", " [nan nan 0.]\n", - " [ 1. 0. 1.]]\n", + " [nan nan 1.]]\n", "\n", "\n", "------------------------------------------\n", "\n", "\n", "Quantum circuit solution\n", - "1101\n", + "0000\n", "\n", "Alice's selection. Column:3\n", - "[[1]\n", + "[[0]\n", " [0]\n", - " [0]]\n", + " [1]]\n", "\n", - "Bob's circuit solution. Row:3\n", - "[1 1 0]\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", "\n", "\n", - "[[nan nan 1.]\n", + "[[ 0. 0. 0.]\n", " [nan nan 0.]\n", - " [ 1. 1. 0.]]\n", + " [nan nan 1.]]\n", "\n", "\n", "------------------------------------------\n", "\n", "\n", "Quantum circuit solution\n", - "0111\n", + "1110\n", "\n", "Alice's selection. Column:3\n", - "[[1]\n", + "[[0]\n", " [1]\n", - " [1]]\n", + " [0]]\n", "\n", - "Bob's circuit solution. Row:3\n", - "[1 0 1]\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", "\n", "\n", - "[[nan nan 1.]\n", + "[[ 1. 1. 0.]\n", " [nan nan 1.]\n", - " [ 1. 0. 1.]]\n", + " [nan nan 0.]]\n", "\n", "\n", "------------------------------------------\n", "\n", "\n", "Quantum circuit solution\n", - "1000\n", + "1001\n", "\n", "Alice's selection. Column:3\n", - "[[0]\n", + "[[1]\n", " [0]\n", - " [1]]\n", + " [0]]\n", "\n", - "Bob's circuit solution. Row:3\n", + "Bob's circuit solution. Row:1\n", "[0 1 1]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", "\n", "\n", - "[[nan nan 0.]\n", + "[[ 0. 1. 1.]\n", " [nan nan 0.]\n", - " [ 0. 1. 1.]]\n", + " [nan nan 0.]]\n", "\n", "\n", "------------------------------------------\n", "\n", "\n", "Quantum circuit solution\n", - "0010\n", + "0101\n", "\n", "Alice's selection. Column:3\n", - "[[0]\n", - " [1]\n", + "[[1]\n", + " [0]\n", " [0]]\n", "\n", - "Bob's circuit solution. Row:3\n", - "[0 0 0]\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", "\n", "Victory!\n", "The following magic square wins the game!\n", "\n", "\n", - "[[nan nan 0.]\n", - " [nan nan 1.]\n", - " [ 0. 0. 0.]]\n", + "[[ 1. 0. 1.]\n", + " [nan nan 0.]\n", + " [nan nan 0.]]\n", "\n", "\n", "------------------------------------------\n", "\n", - "\n", - "100.0%\n" + "\n" + ] + }, + { + "output_type": "error", + "ename": "NameError", + "evalue": "name 'get_state_accuracy' is not defined", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 31\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresults\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 32\u001b[0m \u001b[0mexpected_states\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcheck_result\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresults\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0malpha\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mbeta\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 33\u001b[0;31m \u001b[0mstate_accuracy\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mget_state_accuracy\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresults\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mexpected_states\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m8192\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 34\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mstate_accuracy\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 35\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mNameError\u001b[0m: name 'get_state_accuracy' is not defined" ] } ], @@ -488,22 +530,6 @@ " print(\"please enter values of 1, 2, or 3\")" ] }, - { - "cell_type": "code", - "execution_count": 88, - "metadata": {}, - "outputs": [], - "source": [ - "# function to calculate state accuracy\n", - "def get_state_accuracy(counts, expected_states, shots): \n", - " expected_counts = 0\n", - " for state in counts.keys():\n", - " if state in expected_states:\n", - " expected_counts = expected_counts + counts[state]\n", - " state_accuracy = expected_counts / shots\n", - " return str(state_accuracy*100)+\"%\"" - ] - }, { "source": [ "#### Indeed, using an ideal quantum simulator, a winning probability of 100% is achieved, which exceeds the highest classical winning probability of 88.99% Beyond that, however, it is desirable to quantify the results on a real IBM quantum computer, which will be realized in the last section." @@ -541,27 +567,27 @@ }, { "cell_type": "code", - "execution_count": 38, + "execution_count": 11, "metadata": {}, "outputs": [ { "output_type": "execute_result", "data": { "text/plain": [ - "
" + "
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\n" }, "metadata": {}, - "execution_count": 38 + "execution_count": 11 } ], "source": [ "q = QuantumRegister(4) \n", "c = ClassicalRegister(4) \n", "qc = QuantumCircuit(q, c) \n", - "share_bell_state(qc,0,1,2,3)\n", + "share_bell_state(qc)\n", "qc.draw(output='mpl')\n" ] }, @@ -727,20 +753,20 @@ }, { "cell_type": "code", - "execution_count": 39, + "execution_count": 14, "metadata": {}, "outputs": [ { "output_type": "execute_result", "data": { "text/plain": [ - "
" + "
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\n" }, "metadata": {}, - "execution_count": 39 + "execution_count": 14 } ], "source": [ @@ -748,8 +774,8 @@ "c = ClassicalRegister(4) \n", "qc = QuantumCircuit(q, c) \n", "qc.barrier()\n", - "U(qc,1,0,1) \n", - "V(qc,1,2,3)\n", + "U(qc,1) \n", + "V(qc,1)\n", "qc.draw(output='mpl')" ] }, @@ -863,44 +889,44 @@ }, { "cell_type": "code", - "execution_count": 40, + "execution_count": 16, "metadata": {}, "outputs": [ { "output_type": "execute_result", "data": { "text/plain": [ - "
" + "
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\n" }, "metadata": {}, - "execution_count": 40 + "execution_count": 16 } ], "source": [ "q = QuantumRegister(4) \n", "c = ClassicalRegister(4) \n", "qc = QuantumCircuit(q, c) \n", - "share_bell_state(qc,0,1,2,3)\n", + "share_bell_state(qc)\n", "qc.barrier()\n", - "U(qc,1,0,1) \n", - "V(qc,1,2,3)\n", + "U(qc,1) \n", + "V(qc,1)\n", "qc.measure(q,c)\n", "qc.draw(output='mpl')" ] }, { "cell_type": "code", - "execution_count": 42, + "execution_count": 17, "metadata": {}, "outputs": [ { "output_type": "stream", "name": "stdout", "text": [ - "Accuracy: 100.0%\n" + "Resultant quantum circuit solution:\n\n0000\n\nBring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n\n[0 0 0 0]\n\nCompute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\nCompute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n\n(3,)\nAlice's column solution\n[[0]\n [0]\n [1]]\nWinning conditions fullfilled (Odd quantitude of 1)?: True\n\nBob's row solution\n[0 0 0]\nWinning conditions fullfilled (Even quantitude of 1)?: True\n\nCreate matrix to show results\n\n[[ 0. 0. 0.]\n [ 0. nan nan]\n [ 1. nan nan]]\n\n\n------------------------------------------\n------------------------------------------\n\n\nQuantum circuit solution\n0000\n\nAlice's selection. Column:1\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 0. 0.]\n [ 0. nan nan]\n [ 1. nan nan]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1111\n\nAlice's selection. Column:1\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 1. 0.]\n [ 1. nan nan]\n [ 1. nan nan]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0101\n\nAlice's selection. Column:1\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 0. 1.]\n [ 0. nan nan]\n [ 0. nan nan]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1000\n\nAlice's selection. Column:1\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 1. 1.]\n [ 0. nan nan]\n [ 1. nan nan]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0010\n\nAlice's selection. Column:1\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 0. 0.]\n [ 1. nan nan]\n [ 0. nan nan]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0111\n\nAlice's selection. Column:1\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 0. 1.]\n [ 1. nan nan]\n [ 1. nan nan]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1010\n\nAlice's selection. Column:1\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 1. 1.]\n [ 1. nan nan]\n [ 0. nan nan]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1101\n\nAlice's selection. Column:1\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 1. 0.]\n [ 0. nan nan]\n [ 0. nan nan]]\n\n\n------------------------------------------\n\n\nAccuracy: 100.0%\n" ] } ], @@ -958,14 +984,14 @@ }, { "cell_type": "code", - "execution_count": 43, + "execution_count": 24, "metadata": {}, "outputs": [ { "output_type": "stream", "name": "stdout", "text": [ - "92.88330078125%\n" + "Resultant quantum circuit solution:\n\n0000\n\nBring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n\n[0 0 0 0]\n\nCompute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\nCompute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n\n(3,)\nAlice's column solution\n[[0]\n [0]\n [1]]\nWinning conditions fullfilled (Odd quantitude of 1)?: True\n\nBob's row solution\n[0 0 0]\nWinning conditions fullfilled (Even quantitude of 1)?: True\n\nCreate matrix to show results\n\n[[ 0. 0. 0.]\n [nan nan 0.]\n [nan nan 1.]]\n\n\n------------------------------------------\n------------------------------------------\n\n\nQuantum circuit solution\n0000\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 0. 0.]\n [nan nan 0.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0001\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0010\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 0. 0.]\n [nan nan 1.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0011\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0100\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0101\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 0. 1.]\n [nan nan 0.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0110\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0111\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 0. 1.]\n [nan nan 1.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1000\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1001\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 1. 1.]\n [nan nan 0.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1010\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1011\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 1. 1.]\n [nan nan 1.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1100\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 1. 0.]\n [nan nan 0.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1101\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1110\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 1. 0.]\n [nan nan 1.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1111\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\n\n\n------------------------------------------\n\n\n93.00537109375%\n" ] } ], @@ -986,39 +1012,31 @@ }, { "cell_type": "code", - "execution_count": 44, + "execution_count": 2, "metadata": {}, "outputs": [ { - "output_type": "error", - "ename": "NameError", - "evalue": "name 'IBMQ' is not defined", - "traceback": [ - "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", - "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mIBMQ\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msave_account\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'YOUR_API_TOKEN'\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;31m# retrieve from your IBM-Account: https://quantum-computing.ibm.com\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", - "\u001b[0;31mNameError\u001b[0m: name 'IBMQ' is not defined" + "output_type": "stream", + "name": "stderr", + "text": [ + "configrc.store_credentials:WARNING:2021-05-19 14:15:02,195: Credentials already present. Set overwrite=True to overwrite.\n" ] } ], "source": [ - "IBMQ.save_account('YOUR_API_TOKEN') # retrieve from your IBM-Account: https://quantum-computing.ibm.com" + "IBMQ.save_account('API_Key') # retrieve from your IBM-Account: https://quantum-computing.ibm.com" ] }, { "cell_type": "code", - "execution_count": 45, + "execution_count": 3, "metadata": {}, "outputs": [ { - "output_type": "error", - "ename": "NameError", - "evalue": "name 'IBMQ' is not defined", - "traceback": [ - "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", - "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;31m# load IBM Quantum Experience account\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mIBMQ\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mload_account\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 3\u001b[0m \u001b[0mprovider\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mIBMQ\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_provider\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'ibm-q'\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;31m# get provider\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;31mNameError\u001b[0m: name 'IBMQ' is not defined" + "output_type": "stream", + "name": "stderr", + "text": [ + "/Users/david.drexlin@ibm.com/opt/anaconda3/lib/python3.8/site-packages/qiskit/providers/ibmq/ibmqfactory.py:192: UserWarning: Timestamps in IBMQ backend properties, jobs, and job results are all now in local time instead of UTC.\n warnings.warn('Timestamps in IBMQ backend properties, jobs, and job results '\n" ] } ], @@ -1030,39 +1048,44 @@ }, { "cell_type": "code", - "execution_count": 393, + "execution_count": 69, "metadata": {}, "outputs": [ { - "data": { - "text/plain": [ - "[,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ,\n", - " ]" - ] - }, - "execution_count": 393, - "metadata": {}, - "output_type": "execute_result" + "output_type": "stream", + "name": "stdout", + "text": [ + "Resultant quantum circuit solution:\n\n0000\n\nBring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n\n[0 0 0 0]\n\nCompute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\nCompute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n\n(3,)\nAlice's column solution\n[[0]\n [0]\n [1]]\nWinning conditions fullfilled (Odd quantitude of 1)?: True\n\nBob's row solution\n[0 0 0]\nWinning conditions fullfilled (Even quantitude of 1)?: True\n\nCreate matrix to show results\n\n[[ 0. 0. 0.]\n [nan nan 0.]\n [nan nan 1.]]\n\n\n------------------------------------------\n------------------------------------------\n\n\nQuantum circuit solution\n0000\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 0. 0.]\n [nan nan 0.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0001\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0010\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 0. 0.]\n [nan nan 1.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0011\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0100\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0101\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 0. 1.]\n [nan nan 0.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0110\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0111\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 0. 1.]\n [nan nan 1.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1000\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1001\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 1. 1.]\n [nan nan 0.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1010\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1011\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 1. 1.]\n [nan nan 1.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1100\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 1. 0.]\n [nan nan 0.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1101\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1110\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 1. 0.]\n [nan nan 1.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1111\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\n\n\n------------------------------------------\n\n\n" + ] + } + ], + "source": [ + "job = execute(magicsquare_circuit, least_busy(provider.backends(simulator=False)),optimization_level=3, shots=8192)\n", + "results = job.result().get_counts()\n", + "expected_states = check_result(results, alpha, beta)\n", + "state_accuracy = get_state_accuracy(results,expected_states,8192)" + ] + }, + { + "cell_type": "code", + "execution_count": 70, + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "68.51806640625%\n" + ] } ], "source": [ - "provider.backends()" + "print(state_accuracy)" ] }, { "cell_type": "code", - "execution_count": 394, + "execution_count": 28, "metadata": {}, "outputs": [], "source": [ @@ -1105,8 +1128,786 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "#### As shown above, the MPG circuit outperforms the classical circuit by about 7.8% (88.89% vs. 96.69%) on a real, noisy, quantum computer." + "#### As shown above, the MPG circuit outperforms the classical circuit by about 7.8% (88.89% vs. 96.69%) on a real, noisy, quantum computer. Further optimising can be achieved via measurement error mitigation as can be further seen in this [notebook](https://qiskit.org/textbook/ch-quantum-hardware/measurement-error-mitigation.html)." ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "Resultant quantum circuit solution:\n", + "\n", + "0000\n", + "\n", + "Bring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", + "\n", + "[0 0 0 0]\n", + "\n", + "Compute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\n", + "Compute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n", + "\n", + "(3,)\n", + "Alice's column solution\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "Winning conditions fullfilled (Odd quantitude of 1)?: True\n", + "\n", + "Bob's row solution\n", + "[0 0 0]\n", + "Winning conditions fullfilled (Even quantitude of 1)?: True\n", + "\n", + "Create matrix to show results\n", + "\n", + "[[ 0. 0. 0.]\n", + " [nan nan 0.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0000\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 0. 0.]\n", + " [nan nan 0.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0001\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0010\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 0. 0.]\n", + " [nan nan 1.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0011\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0100\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0101\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 0. 1.]\n", + " [nan nan 0.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0110\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0111\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 0. 1.]\n", + " [nan nan 1.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1000\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1001\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 1. 1.]\n", + " [nan nan 0.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1010\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1011\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 1. 1.]\n", + " [nan nan 1.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1100\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 1. 0.]\n", + " [nan nan 0.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1101\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1110\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 1. 0.]\n", + " [nan nan 1.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1111\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Resultant quantum circuit solution:\n", + "\n", + "0000\n", + "\n", + "Bring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", + "\n", + "[0 0 0 0]\n", + "\n", + "Compute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\n", + "Compute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n", + "\n", + "(3,)\n", + "Alice's column solution\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "Winning conditions fullfilled (Odd quantitude of 1)?: True\n", + "\n", + "Bob's row solution\n", + "[0 0 0]\n", + "Winning conditions fullfilled (Even quantitude of 1)?: True\n", + "\n", + "Create matrix to show results\n", + "\n", + "[[ 0. 0. 0.]\n", + " [nan nan 0.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0000\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 0. 0.]\n", + " [nan nan 0.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0001\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0010\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 0. 0.]\n", + " [nan nan 1.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0011\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0100\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0101\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 0. 1.]\n", + " [nan nan 0.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0110\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0111\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 0. 1.]\n", + " [nan nan 1.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1000\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1001\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 1. 1.]\n", + " [nan nan 0.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1010\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1011\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 1. 1.]\n", + " [nan nan 1.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1100\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 1. 0.]\n", + " [nan nan 0.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1101\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1110\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 1. 0.]\n", + " [nan nan 1.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1111\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "State Accuracy \n", + "noisy: 83.8134765625%\n", + "mitigated: 87.33150194452158%\n" + ] + }, + { + "output_type": "execute_result", + "data": { + "text/plain": [ + "
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/sry8fBLqEMi0OmBFTU3NBU016PDCxswuBb4KDAVWAle4+7NNtO0J3A5MAY4A/uHusxu0OR24GJgM9ARWAd9z98fS9R5ERKTrmDp16hbgY5nOQ1qnQytPM5sD3Ax8n1CIPAf81cwOaWKXHOBd4GfA4020mQX8HTgtivkX4I9mdkI7pi4iIiJZoENnHjazJcA/3f3ChHVrgIfd/eoW9v0ZMKlhj00Tbf8PeNbdv9JcO808LCIiDTWceViyS4ddijKzHsBU4McNNs0Djmvnw/UG2vSsCRGRzmDt+Oua3DZ2ddPbRLqqjhxjM5BwaamiwfoK4EPtdRAz+yIwArinie0XARcBDBs2jPnz5wMwZswYevfuTVlZGQADBgxg4sSJLFy4EAhPO50xYwbLli2jsrISgJKSEioqKti4cSMA48aNIy8vjxUrwmDtwYMHc9hhh7Fo0SIA8vLymD59OqWlpezevRuAadOmsWnTJjZv3gzA+PHjycnJYdWqVQAUFhYyevRoFi9eDECvXr2YNm0aS5YsoaqqCoDp06ezfv16ysvLAZgwYQK1tbWsXr0agOHDhzNixAiWLFkCQEFBASUlJSxevJjq6moAZsyYwauvvsqWLVsAmDRpEtXV1axZswaAkSNHMmTIEOp7uPr06cOUKVNYtGgRNTU1AMycOZOVK1eyfft2AIqLi9m1axfr1q0DoKioiP79+7Ns2TIA+vXrR3FxMQsWLMDdMTNmzZpFWVkZO3eGunTKlCns2LGDDRs26Puk71OX/D7l07TKykp9n9LwfZLs1mGXosxsGLAZmJk4WNjMrgU+4+6Ht7B/i5eizOxThILm060ZPKxLUSLS2anHpuPpUlR268jBw9uAWqCwwfrBHNiLk7SEouZc3RElIiLSNXVYYePu+4ClwEkNNp1EuDsqZWZ2JnAvcJ67P9yWWCIiIpK9OnoemxuBe6K7lv5BmH9mGGGuGszseuBYd/9g/Q5mNgHoQRijU2BmRwO4+4vR9k8TemquBBaaWX2P0D5339EB70mkU2ruEgboMoaIxFOHFjbu/oCZDQC+SZigbwVwqru/HjUZCoxtsNtfgFEJr5dH/9ZPbX0x4X3cFC31FgCz2yl1ERERyQIdPvOwu/8c+HkT285rZF1RC/Fmt0deItI5aLCsiLSFnnkhIiIisaHCRkRERGJDhY2IiIjERoePsRERkfddeFPz26/qkCxE4kM9NiIiIhIbKmxEREQkNnQpSkQ6lC69iEg6qcdGREREYkOFjYiIiMSGChsRERGJDRU2IiIiEhsqbERERCQ2dFeUSBZr7g4j3V0kIl2RChtpVHNPWAY9ZVlEmqafH5JJuhQlIiIisaHCRkRERGJDhY2IiIjEhsbYdFGa1l5EROJIPTYiIiISGypsREREJDZU2IiIiEhsaIyNSCtoXg4RkeygHhsRERGJDRU2IiIiEhsqbERERCQ2VNiIiIhIbKiwERERkdjQXVEiaCZmEZG4UI+NiIiIxIZ6bEREJCnq4ZTOTIWNiEgbaQJHkc5Dl6JEREQkNlTYiIiISGyosBEREZHYUGEjIiIisaHCRkRERGJDhY2IiIjEhgobERERiQ3NY9MOWpqs6s4rOiILERERUY+NiIiIxIYKGxEREYkNXYrqAM1Nt66p1kU6Pz0bSSR7qMdGREREYiOpwsbMuplZt4TXhWZ2gZkd3/6piYiIiCQn2R6bx4HLAMysACgFfgTMN7Nz2zk3ERERkaQkW9hMBf4efX06UAkMBi4ErmxNADO71MzWm9m7ZrbUzE5oof2RZrbAzKrMbLOZXWNm1qDNF83s5ajNahVZIiIiXVOyhU1v4O3o65OBP7r7fkKxM7alnc1sDnAz8H1gMvAc8FczO6SJ9n2AJ4EK4BjgcuCrwNyENpcAPwS+A0wErgVuNbOPJvneREREJMslW9i8ARxvZgcBpxCKDoD+wN5W7D8XuNvd73T3l939MuAt4JIm2p8F5AOfc/cV7v4HQhEzN6HX5hzgTnf/vbuvc/f7gV8AX0vyvYmIiEiWS7awuRG4B9gEbAYWRutnAi81t6OZ9SBcyprXYNM84LgmdpsOPOvuVQnr/gYMA4qi13nAuw32qwKONbPuzeUkIiIi8ZLUPDbufoeZLQVGAk+6e120aS3wrRZ2HwjkEC4rJaoAPtTEPoWEIqph+/pt6wmFzvlm9ghhMPNU4AKge3TMtxJ3NrOLgIsAhg0bxvz58wEYM2YMvXv3pqysDIABAwYwceJEFi4MtVtubi4zZsxg2bJlVFZWAlBSUkJFRQWtuArXpKqqKjZt2sTmzZsBGD9+PDk5OaxatSq8ycJCRo8ezeLFiwHo1asX06ZNY8mSJVRVhXpv+vTprF+/nvLycgAmTJhAbW0tq1evBmD48OGMGDGCJUuWAFBQUACUpJwzwLJly5gyZQqLFi2ipqYGgJkzZ7Jy5Uq2b98OQHFxMbt27WLdunUAFBUV0b9/f5YtWwZAv379KC4uZsGCBbg7ZsasWbMoKytj586dAEyZMoUdO3awYcMGoG3fp40bNwIwbtw48vLyWLFiBQCDBw8GJrTpfLzyyitp+T6VlJSwePFiqqurAZgxYwavvvoqW7ZsiY48O+Wc6//vd/z3KfWca2pqWLRoEQB5eXlMnz6d0tJSdu/eDcC0adPS8nkKQwlTt23bNqqrq1mzZg0AI0eOZMiQIZSWltKW81FZWZmRzxMUpJwzZO7nXkufp0mTJr23TbKXuXvHHMhsGKGXZ6a7P5uw/lrgM+5+eCP7zAM2uvv5CetGARuA6e7+vJn1Am4lXJIyQuFzL/D/gCHuvqVh3HolJSUefrC0TYuTd912XZPbMjVBX1tyhvhNLJit56O5vLMxZ+h6nxedj87HzJa6e9v++pOMSXqCvuiuppVmttfMxkTrvmZmZ7aw6zagltDTkmgwB/bi1Ctvoj31+7h7lbt/gTAWpwg4hFD47IqOKSIiIl1EshP0XQF8kzA4N/GW6zeBLzW3r7vvA5YCJzXYdBLh7qjGLAZOMLOeDdq/SSheEuPvd/dN7l4LfBr4c8KlMhEREekCku2xuRi40N1vBmoS1i8j3GrdkhuB86LZio8ws5sJA4FvBzCz683s6YT29xHutrrbzCaZ2emEx7Lc6NE1NDM7zMzOMbNxZnasmd0PTAK+nuR7ExERkSyX7EMwRwErGlm/H+jV0s7u/oCZDSD0+gyNYp3q7q9HTYaSMBLX3d8xs5MIY2hKgZ3ADYQCqV4O4Tby8VEezwDHufuGpN6ZiIiIZL1kC5t1wBTg9QbrTwVWtSaAu/8c+HkT285rZN1LhNvJm4r3MmGyPxEREeniki1sfgz8zMzyCWNsppvZOYQ7kL7Q3smJiIiIJCPZeWx+bWa5hEci5BMm69sMXO7uD6QhPxEREZFWS7bHBne/E7jTzAYC3ZqbJ0ZERESkIyVd2NRzd80RIyIiIp1Ki4WNmf0TmOXuO83sJaDJqYrd/aj2TE5EREQkGa3psfkDUJ3wdcc8g0FEREQkSS0WNu7+7YSvr0trNiIiIiJtkOwjFf5uZgc3sr6Pmf293bISERERSUGyj1SYDfRoZH1P4IQ2ZyMiIiLSBq26K8rMpiS8PMrMdiS8zgFOIcxnIyIiIpIxrb3du5QwaNiBeY1srwIua6+kRERERFLR2sJmNOERCuuAY4GtCdv2AVvcvbadcxMRERFJSqsKm4Snbyc7JkdERESkw7Rmgr7Tgf919/3R101y90faLTMRERGRJLWmx+ZhoBDYEn3dFCcMJBYRERHJiNZM0Netsa9FREREOhsVKiIiIhIbrR1j0yoaYyMiIiKZ1NoxNq2hMTYiIiKSUUmNsRERERHpzFS0iIiISGxoHhsRERGJDc1jIyIiIrGheWxEREQkNlSoiIiISGwkXdiY2RQz+62ZlUbLPWY2JR3JiYiIiCQjqcLGzM4CXgCGAn+JliHA/5nZ2e2fnoiIiEjrtWbwcKLvAd9y9+8nrjSzq4HvAve2V2IiIiIiyUr2UtQg4MFG1j8EDG57OiIiIiKpS7aweQaY3cj62cCCtiYjIiIi0hbJPgTzr8D1ZlYCPB+t+wBwOnBdu2cnIiIikoRUH4J5UbQkugX4eZszEhEREUmRHoIpIiIisaGiRURERGIj2du9MbP+wL8DhwA9Ere5+3faKS8RERGRpCVV2JjZB4DHgWrCrd+bCZP1VQMbABU2IiIikjHJXor6EfA7YDjwLnAioeemFPhh+6YmIiIikpxkC5ujgJ+5uwO1QJ67VwBfQ7d7i4iISIYlW9jsS/i6AhgVfb0bGNYuGYmIiIikKNnBw8uAY4BXgfnAd81sCHA28M/2TU1EREQkOcn22HwDeDP6+pvAVsLEfP04cMI+ERERkQ6VVI+Nu5cmfL0V+HC7ZyQiIiKSoqTnsQEws7HAEdHLVe6+rv1SEhEREUlNsvPYDADuAj4G1L2/2v4MfMHdt7dzfiIiIiKtluwYm18ChwInAD2jZSYwGrizfVMTERERSU6yl6JOAT7o7osT1v3DzP4TeKr90hIRERFJXrI9NluBPY2s3wu06jKUmV1qZuvN7F0zW2pmJ7TQ/kgzW2BmVWa22cyuMTNr0KaHmX0nilttZm+Y2eWtflciIiISC8n22HwHuMnMznH3zQBmNhy4gVY8J8rM5gA3A5cCi6J//2pmE9z9jUba9wGeBBYS5s8ZD9xNKK5uSGj6e2Ak4ZbzNcAQoFeS701ERESyXIuFjZm9BHjCqtHABjPbHL2uf27UYMIYnObMBe529/rxOJeZ2b8DlwBXN9L+LCAf+Jy7VwErzOwIYK6Z3ejubmYnAx8Cxrr7tmi/DS29LxEREYmf1vTYPNweBzKzHsBU4McNNs0Djmtit+nAs1FRU+9vwH8DRcB64BPAC4Ri51ygCvgr8HV3390euYuIiEh2aLGwcfdvt9OxBgI5hGdMJaog9Lg0phDY1Ej7+m3rgTHADKAa+BRwMGE25GHAGQ0DmtlFRLMkDxs2jPnz5wMwZswYevfuTVlZGQADBgxg4sSJLFy4EIDc3FxmzJjBsmXLqKysBKCkpISKigpgbItvvilVVVVs2rSJzZtDB9j48ePJyclh1apV4U0WFjJ69GgWLw7jtXv16sW0adNYsmQJVVWh3ps+fTrr16+nvLwcgAkTJlBbW8vq1asBGD58OCNGjGDJkiUAFBQUACUp5wywbNkypkyZwqJFi6ipqQFg5syZrFy5ku3bw3Cr4uJidu3axbp1YZqjoqIi+vfvz7JlywDo168fxcXFLFiwAHfHzJg1axZlZWXs3LkTgClTprBjxw42bNgAtO37tHHjRgDGjRtHXl4eK1asAGDw4MHAhDadj1deeSUt36eSkhIWL15MdXU1ADNmzODVV19ly5Yt0ZFnp5xz/f/9jv8+pZ5zTU0NixYtAiAvL4/p06dTWlrK7t3hb5hp06al5fMUOqRTt23bNqqrq1mzZg0AI0eOZMiQIZSWltKW81FZWZmRzxMUpJwzZO7nXkufp0mTJr23TbKXhQd1J7mT2YmE3wQOrHT3+a3YZxiwGZjp7s8mrL8W+Iy7H97IPvOAje5+fsK6UYRLTdPd/fmozQlAobu/E7U5mdCzUxg9fbxRJSUlHn6wtM2FNzW//arbrmty29jVTW9Lp7bkDJnLO12y9Xw0l3c25gxd7/Oi89H5mNlSd2/bX3+SMclO0Dcc+CPhklL9M6OGmVkp8El3f7PJnWEbUEvoaUk0mAN7ceqVN9GehH3eAjbXFzWRl6N/D2kmtoiIiMRMsrd7/5RQnBzq7iPdfSQwLlr30+Z2dPd9wFLgpAabTgKea2K3xcAJZtazQfs3eX+A8D8IxVVi3+hh0b+vN/tuREREJFaSLWxOAr7o7uvrV0TPibqcAwuWxtwInGdmF5jZEWZ2M2EszO0AZna9mT2d0P4+whw5d5vZJDM7HbgKuNHfv4Z2H2EOnV+b2UQzO55wS/nD7r4FERER6TJSeghmI+pabgLu/kD0vKlvAkOBFcCp7l7fszKUhJG47v6OmZ0E3AqUAjsJ89fcmNBmt5l9iDBg+IWozaOEAkhERES6kGQLm6eBn5rZZ9x9I4CZHULoIXm62T0j7v5z4OdNbDuvkXUvEZ5H1VzM1cDJrTm+iIiIxFeyl6IuJ0yYt87MXjezDcDaaJ0eYSAiIiIZlWyPzXbgWODfgMMBA1a5ux6AKSIiIhnX6sLGzHKAd4Bid3+S8AwnERERkU6j1Zei3L2WcPt0j/SlIyIiIpK6ZMfY/DfwAzMbmI5kRERERNoi2TE2VxKe7r3ZzDYBexI3uvtR7ZWYiIiISLKSLWweJjwfytKQi4iIiEibtKqwMbN84EfAJ4DuhDlrLnP3belLTURERCQ5rR1j823gPOBx4PfAh4Db0pSTiIiISEpaeynqdOB8d78fwMx+B/zDzHKiu6VEREREMq61PTYjgWfrX7j7/wE1hAdYioiIiHQKrS1scoB9DdbV0H4P0RQRERFps9YWJgbca2bVCet6Anea2d76Fe7+sfZMTkRERCQZrS1sftPIunvbMxERERGRtmpVYePun093IiIiIiJtlewjFUREREQ6LRU2IiIiEhsqbERERCQ2VNiIiIhIbKiwERERkdhQYSMiIiKxocJGREREYkOFjYiIiMSGChsRERGJDRU2IiIiEhsqbERERCQ2VNiIiIhIbKiwERERkdhQYSMiIiKxocJGREREYkOFjYiIiMSGChsRERGJDRU2IiIiEhsqbERERCQ2VNiIiIhIbKiwERERkdhQYSMiIiKxocJGREREYkOFjYiIiMSGChsRERGJDRU2IiIiEhsqbERERCQ2VNiIiIhIbKiwERERkdhQYSMiIiKx0eGFjZldambrzexdM1tqZie00P5IM1tgZlVmttnMrjEzS9g+y8yeM7PtUZtXzOzK9L8TERER6WxyO/JgZjYHuBm4FFgU/ftXM5vg7m800r4P8CSwEDgGGA/cDewBboia7QZ+CrwE7AWOB+4ws73u/vO0viERERHpVDq6x2YucLe73+nuL7v7ZcBbwCVNtD8LyAc+5+4r3P0PwA+BufW9Nu6+1N3vd/eV7r7e3e8F/gY02xMkIiIi8dNhhY2Z9QCmAvMabJoHHNfEbtOBZ929KmHd34BhQFETx5kcxVvQlnxFREQk+3TkpaiBQA5Q0WB9BfChJvYpBDY10r5+2/r6lWa2CRhEeE/fdvfbGwtoZhcBFwEMGzaM+fPnAzBmzBh69+5NWVkZAAMGDGDixIksXLgQgNzcXGbMmMGyZcuorKwEoKSkhIqKCmBss2+8OVVVVWzatInNmzcDMH78eHJycli1alV4k4WFjB49msWLFwPQq1cvpk2bxpIlS6iqCvXe9OnTWb9+PeXl5QBMmDCB2tpaVq9eDcDw4cMZMWIES5YsAaCgoAAoSTlngGXLljFlyhQWLVpETU0NADNnzmTlypVs374dgOLiYnbt2sW6desAKCoqon///ixbtgyAfv36UVxczIIFC3B3zIxZs2ZRVlbGzp07AZgyZQo7duxgw4YNQNu+Txs3bgRg3Lhx5OXlsWLFCgAGDx4MTGjT+XjllVfS8n0qKSlh8eLFVFdXAzBjxgxeffVVtmzZEh15dso51//f7/jvU+o519TUsGjRIgDy8vKYPn06paWl7N69G4Bp06al5fMEg1POGWDbtm1UV1ezZs0aAEaOHMmQIUMoLS2lLeejsrIyI58nKEg5Z8jcz72WPk+TJk16b5tkL3P3jjmQ2TBgMzDT3Z9NWH8t8Bl3P7yRfeYBG939/IR1o4ANwHR3fz5h/WjCp+0DhMtVX3b3e5rLqaSkxMMPlra58Kbmt19123VNbhu7uult6dSWnCFzeadLtp6P5vLOxpyh631edD46HzNb6u5t++tPMqYje2y2AbWEnpZEgzmwF6deeRPtabiPu9f33rxkZkOA64BmCxsRERGJlw4bY+Pu+4ClwEkNNp0EPNfEbouBE8ysZ4P2bxJ6bZrSDchLLVMRERHJVh19V9SNwHlmdoGZHWFmNxMGAt8OYGbXm9nTCe3vI9zCfbeZTTKz04GrgBs9uoZmZpeZ2UfMbFy0nA9cCdzbkW9MREREMq9D57Fx9wfMbADwTWAosAI41d1fj5oMJWEkrru/Y2YnAbcCpcBOwvw1NyaEzSGMqSkCaoC1hOKn0cHDIiIiEl8dWtgARJPmNTpxnruf18i6l4CZzcS7CbipfbITERGRbKZnRYmIiEhsqLARERGR2FBhIyIiIrGhwkZERERiQ4WNiIiIxIYKGxEREYmNDr/dW6QtmntGzZ1XdFQWyWnpuTqdNW8RkWykHhsRERGJDRU2IiIiEhu6FCXtTpdeREQkU9RjIyIiIrGhwkZERERiQ5eiOrlsvAtIRDqH5n5+XHXbdc3uO3Z189szZe3465rc1llzlo6lwkZERDqNlsboXdUhWUg206UoERERiQ0VNiIiIhIbuhQlkmEaM9B+NCZNRNRjIyIiIrGhHhsROYAmWRSRbKXCRmKjuUs6oMs6IiJdgS5FiYiISGyox0Y6nAbLiohIuqjHRkRERGJDhY2IiIjEhi5FZTENlhUREflXKmxEpEvQHwIiXYMuRYmIiEhsqLARERGR2FBhIyIiIrGhwkZERERiQ4OHRSRpGogrIp2VemxEREQkNlTYiIiISGyosBEREZHYUGEjIiIisaHCRkRERGJDhY2IiIjEhgobERERiQ0VNiIiIhIbKmxEREQkNlTYiIiISGyosBEREZHYUGEjIiIisaHCRkRERGJDhY2IiIjERocXNmZ2qZmtN7N3zWypmZ3QQvsjzWyBmVWZ2WYzu8bMrEGbWVGsd81snZldnN53ISIiIp1RhxY2ZjYHuBn4PjAZeA74q5kd0kT7PsCTQAVwDHA58FVgbkKb0cBfoliTgeuBW8zsU+l7JyIiItIZdXSPzVzgbne/091fdvfLgLeAS5pofxaQD3zO3Ve4+x+AHwJzE3ptLgbedPfLoph3Ar8BrkzvWxEREZHOpsMKGzPrAUwF5jXYNA84rondpgPPuntVwrq/AcOAooQ2DWP+DSgxs+5tyVlERESyS0f22AwEcgiXlRJVAIVN7FPYRPv6bc21yY2OKSIiIl2EuXvHHMhsGLAZmOnuzyasvxb4jLsf3sg+84CN7n5+wrpRwAZgurs/b2avAve4+38ntJkFzAeGunt5g5gXARdFL8cDq9vnHTZrILAti+Jma+xszDmdsbMx53TGzsac0xk7G3NOd+x6o9x9UJqPIWmS24HH2gbUcmDvzGAO7HGpV95EexL2aapNDbC9YUB3/wXwi9al3D7MrNTdS7IlbrbGzsac0xk7G3NOZ+xszDmdsbMx53THlnjosEtR7r4PWAqc1GDTSYQ7mhqzGDjBzHo2aP8modemvs2HGolZ6u7725KziIiIZJeOvivqRuA8M7vAzI4ws5sJA4FvBzCz683s6YT29wF7gbvNbJKZnQ5cBdzo719Dux0YYWY3RTEvAM4DftxB70lEREQ6iY68FIW7P2BmA4BvAkOBFcCp7v561GQoMDah/TtmdhJwK1AK7ARuIBRI9W3Wm9mpwE8It42/CVwe3RreWaTr0lc6L6llY+xszDmdsbMx53TGzsac0xk7G3NOd2yJgQ4bPCwiIiKSbnpWlIiIiMSGChsRERGJDRU2IiIiEhsqbNLMzLLyHGdj3spZREQ0eDjNood1jgJ2ER7oudPdd7dT7Fyg1tPwTUxX3tmYcxQ7LXlnY84iIp2ZCps0MrMZwAXAR4ACoAx4HlgILHT3rWbWzd3r2nicHAB3r21jyvXx0p53NuYcHafd8s7GnBNiDgV6A1WEZ8CVu/u77RQ7191r2iNWI7HTknc25hzFTkve2ZizxIcKmzQys1XAGuC3wA7gY8CJwBDgGeBKd99sZpbsX9Vm9gThF+Av3H1bwvpcoM7d68ysN/BusjMwpyvvbMw5nXlnY85RjEuBLwCTgP2EGcUXA38nFGTVqeTcyHHau/hNe97ZmHN0nPYs2LMuZ4kZd9eShgWYCWwBchrZ9jHgJcIEhUNTiH08UEeYjLCO8Ff+54gK1ahNL+AB4NjOkHc25pzOvLMx52i/2YSH2f4QOAL4MHAnoUDbDHwPyE3xM3M8odfqAqBHg225hDGBBvRPfC+ZzDsbc05n3tmYs5b4LRlPIK4L8FlgFXBE9LoX0D1h+1jgNeALKcS+BngcmAacEf2C2gnsAx4FPghMjX6p9e4MeWdjzunMOxtzjmLfR+gFari+O3AxoVi7K8XPzG8ID8p9i/AQ2yeAjzRoc3y0PqlfjunKOxtzTmfe2ZizlvgtuiMjfR6P/r0QwN2r3H2/meWYWY67ryV0z34ghdg1wBvAP9394egYM4ErgD7AY8ALwOPuvquT5J2NOacz72zMGUJx1K/+wbRm1jMa87Df3W8HrgaON7OJKeRdRHjcysnAf0brHjKzSjO7y8yKgU8Dwzz5MRbpyjsbc05n3tmYs8RNpiurOC/A54FKwl/eXwdGJ2w7ivDL58wU4uYDRzeyvhswgPDBryM8hyuZuPVjrs4D3mnPvNOVczae63Se53Sfa+AUYFvDvIj+Qo6O/QYwK8m4w4BfAv8Zvc4B+gHHEB58+wJQHeX90c6QdzbmnO68szFnLfFbMp5A3BfgSOAOwpiJcuAV4G/ABuB/U4hnDV53iz7kieMnPg7UtDHvYuDnwIuErt825d2K47VHzll3rjv6PLclZ8L4hZ6EB87WEMbuXAwMiLYPJAwarUwxrwHAyEbW5wKDgO8Ab7ch7xsJg1nbLe8OyDlrznU6z3M6z7WW+C26KyqNzKw74YdSDuGX7mRgHDAcmAc85u6VKcbOBdwT7giI5kQBuBQY4e5XpxD3vbsVzKwf4c6Gw4AxwAjgybbk3cxxv0iKOUf7Z9W5ztR5jo7XpnMdxTiVMCD5aMIvnArC+c8DfunuP25jjgfc8WJmjxLm5flUG+J+hDC+6WhgMO2cdyPHe5S255x157qjz3N0zEdp47mWeFBh087MrD/wUcIPom3A68A/gfnuvrEdY28ldOmuBha5+ysJ7Qzo5e57UzxOm2/FTOGY3YCeyeSc7ec6E+c5Om7S57rB/ge5+x4zyyf0OB1OKMjygF8Dr3qKt+CaWW9vMO4nOscHAbcAN7v7iynETSwkhwITgEOA0YRehjbl3cQxC9qScxQjq851Js5zdKw2n2uJDxU27czMHiH88FlKGKg5mHDHSwXwIPAbT3FgWxOx86LYfwB+5SlO5mZmnwSWuvsbCeu6AdTHNLM8d69ua9xG2nT3JOdSifbLunOdrvPcVOxG2qR6ro8A5hJ6wtYSisjFwLOeME9OKhrEfo1QRL4Yxd6Y0C6l85KwfyYK9lQ+M1l9rjNYsLfp/4fESKavhcVpIczbsAc4KmFdAfBJ4H7CLJx3AD1Ifk6LlmLvBW5PMfZ4wqC7KuApQhdy3wZtcgjzRxye6bjZeq7TfD7SGXssobdqIXA98EfC+IkXgYeBk9rwmWks9mJgGaE4PbkNsYcA5wD9G6zvxvt/1OUAee0Rt5F2PaN/k/k/knXnOl3nOd3nWkt8l4wnEKeFcAvi87x/B0Bug+2nEeZxSGpStA6IfRXwXPQD5CHCc4veBu4hzISbCxwa/eI8JNNxs/Vcp/l8pDP2bcD/kjDnTfQL5/PRL8k9wPkpfmbSGfuW6P3uiM7JqTT45Uq4THJlw/WZiJut5zrN5yNtsbXEd8l4AnFagBLCnS1fSFjXnff/muhBmDzqxk4W+/roh17f6HUR4Zf7QsLdDesIl2Re6gxxs/Vcp/l8pDP2X4BvR1/n0GC2ZOAGQiGY38liL47Oy+cJvVjVhNmYbwGmRG3+G3itM8TN1nOd5vORttha4rtkPIG4LYTpw98BrqWRWV0JXcqXdZbYhO7i44GzGtnWg3C3znWEv5o+n+m42Xqu03k+0n2ugS8TxmIc3iBuj+jrCcB64N86S2zCvCcPARdFr3MJY6a+Fn3vagnTAuwBvpzpuNl6rtN5PtJ9rrXEd8l4AnFcgP9HeC7KPsJfGV8ELoq+fpUU/trqiNhR/AOmIifMWlsHHNQJ414JbEzTuU5n7APGArTH+UhHbMIdLcujX3oHPOKBUDjtS+V8pCs24c6ejwLTGtmWDxxLGFdSQ7irLaNxs/Vcp/N8pPtca4nvorui0sTMRhEeAHcq4Zk97xLmJrnb3Rd11tgJx8ghPAXazexKwkyhH+0scc2sm79/F9EE4ATCrKfTCT+cUz4f6Ygd3UZr3sydVKmej3THjr5XfQgPNvws4TLGPEKBNwmYAbzo7ud2ltgNjwN08wa3GJvZ3cBYdz+hk8XtTbj8chbhEmi7nY90xo7iH3BHVFvPR0fElnhRYdNOol+AE4C+hIGbz7n7poTt/QizYiZ9wjsw9l7geXdf36DNkcDuhuszEbeZ43UjdKm/G52PXd5Oz4tJZ+wGxykmzMra5vORrthmNolQVH+QMPnaOuBe4BF3L++ssaP47xUiZtYL+BNwm7v/sTPEbRCnJ2GiyZmEwd9TCD0tKZ2PdMWOPhve1M+eNp6PtMWWeFNh0w7M7CrCX0DjCAPbtgNOeH7J7wm/1GsSewI6YezNUew6Qnf1fcA/ko2ZzrhR7FHAdnff3cT2lOfQSFfsluK2RZpjdyM8fmEQoet/A7DQ3XcktOnr7u908tibgQXuviWhTS4w1d2XZDpuC8dM7OFM6XxkInbCMboDJe6+OJtiS3ZTYdNGZjaA8EP5q+5+u5mNJFz7nU64u6YncJW7z8+i2FOj2Fe7+/xkiqZ0xY1i9yM8S+kfhEnyngXe8gYTzpnZDGCNu1dkOnaScde6+1tpyjnZ2L2Bu4B/IxSlm6NNe4G/A79395ejtkkVfB0cexPhD4EqYAFwryfMHJ3puFHs7oTxL697IxPMtbFYT0vsluK2RTpjSxfhnWCgTzYvhIe8vdDEtmLCpFqVwJiuEDvNOX+JMJ/FPMJtn1sIv2xOIfwV3Q0YSRjYm+wkdGmJnY05R7G/AZQBx0SvDwfOJkxMWEqYD2VQip+ZTMV+AXgsldjpihvFuoJwZ8+vCYNlCznwVuw+hMt03TtD7CTinkZ011VniK2laywZTyDbF+A/CNenZ0av/2V+CEIPxfPApV0hdppzvpUwm3A3oDfhLqVSwl/QrxLms7iB1J6mnJbY2ZhzFPtZYG4j63MIYzPWAE+k+JnJuthpznkx8Ex0jNro83MjYTBv36jNxYTLzp0idjbmrKXrLBlPINsXwrX2+YRBbEc20WYR8K2uEDuNcbsT/kL+Lw786+1Q4LvRL5c64JudIXY25hztn0uY7O8fRL0QHFigfhBYARTHPXaacx5EmETxs9HrEcA3CYVpHWFSxa8RLjne3BliZ2POWrrWkvEEsnnh/TFKxxOet7Kf0CX9ccI14qnAZcBOoCjusdOZcxS3O9Av+jonep34y+Vwwl94IzpL7GzMOdr3A4QHMP4QGNLI9pHAbmB4V4idxrhDCcXpKY1sm0zolasffN8pYmdjzlq61pLxBOKwEK73DgL+Hfgd4Rk9VYQu1DW0YVbMbIydjri8XzSNBQY32NYt+vdbwBudJXY25ly/P6GX4kJgK6EQvRP4EOFhm/8B/JYmxlLFLXY6c47i9yLhIY71S8L27wHLO1PsbMxZS9dZdFdUisxsMOGBg3MJAzjfJdyO/Tjhzo6DCQ9n+4cncXdOtsbuwJy3EGYafYsw3foj7r4nancaYV6cBZmOnY05N3Gsg4HzCJPnHU2YR6ka+D/gem/Dbc3ZGDuNcRu9O8nM8gm9n7929x92ptjZmLN0DSpsUhTNeDmRcAfHDqA/4a6fwwm/0L/u7i90ldgZyHlyFHsT8CN3n9eZYmdjzlHsPoQJCD1hXTfCoO8Cwgy1e1L5BZ6NsTs650ba9ATmEG6B35fp2NmYs3RBme4yysaF0DW6m+jOn4R1o4AzCbffvgZM7gqxM5TzIYRLAPOA1Z0pdjbmnBDrDuB8wsy0fZpoUz+u54BnUsUtdifI+eA0fh+Tjp2NOWvpekvGE8jGhfDX8kvAB5rY3oNw2+31XSF2Nuas89Hovp8hDMp8m/A4gzuA0wl3WfWK2hQAj9LEHW9xip2BnD9JGDNVH7v+kQGTOkPsbMxZS9dcMp5ANi7RB+xpwnwL44gGazZocxnhgXKxj52NOet8NBr7TsKEc2MITzZ/iTAb8HLCgM0TgUuAfV0hdjbmrPOhRYsKm9RPXLj988XoF8x5hNs9D4q25RNm2L23q8TOxpx1Pv4lZi7wdeAHDdZPBG4ijN3ZRhiofFfcY2djzjofWrSEJeMJZPNCGDj4AOFW5m2EwZy/At4AlpBk93S2x87GnHU+/iVmP6JHLxAuaVmD7XMIlwuO7gqxszFnnQ8tWnS7d7uIbr89DfgE4TbnFcBDnuJD8bI9djbmnM7Y2ZhzQvxuhF8ytWZ2IWG21/yuGjsbc05n7GzMWeJPhU07S/aJ1XGPnY05pzN2NuacEH8uYWbjHyl2duacztjZmLPEkwobEWkVM+sO1Kap4Mu62NmYczpjZ2POEk8qbERERCQ2umU6AREREZH2osJGREREYkOFjYiIiMSGChsRERGJDRU2IiIiEhsqbERERCQ2/j97M6YYovT+dwAAAABJRU5ErkJggg==\n" + }, + "metadata": {}, + "execution_count": 32 + } + ], + "source": [ + "# measurement error mitigation on backend 1\n", + "results_backend1_mem = []\n", + "\n", + "# normal execution\n", + "noisy_counts = execute(magicsquare_circuit, backend=backend, shots=8192, optimization_level=0).result().get_counts() # get results\n", + "expected_states = check_result(noisy_counts, alpha, beta)\n", + "state_accuracy = get_state_accuracy(noisy_counts,expected_states,8192)\n", + "results_backend1_mem.append((\"noisy\", noisy_counts, state_accuracy))\n", + "\n", + "# measurement error mitigation\n", + "qr = QuantumRegister(4)\n", + "meas_calibs, state_labels = complete_meas_cal(qr=qr, circlabel='mcal') # setup calibration measures on basis states\n", + "\n", + "cal_results = execute(meas_calibs, backend=backend, shots=8192).result() # execute calibration measures\n", + "meas_fitter = CompleteMeasFitter(cal_results, state_labels, circlabel='mcal')\n", + "meas_filter = meas_fitter.filter # get the filter object from calibration\n", + "\n", + "mitigated_counts = meas_filter.apply(noisy_counts) # apply filter to get mitigated results\n", + "expected_states = check_result(mitigated_counts, alpha, beta)\n", + "state_accuracy = get_state_accuracy(mitigated_counts,expected_states,8192)\n", + "results_backend1_mem.append((\"mitigated\", mitigated_counts, state_accuracy))\n", + "\n", + "# compare state accuracies\n", + "print(\"State Accuracy \\n\"+\n", + " \"\\n\".join([str(label)+': '+str(state_accuracy) for label, counts, state_accuracy in results_backend1_mem]))\n", + "\n", + "# plot results\n", + "plot_histogram([counts for optim_level, counts, state_accuracy in results_backend1_mem],\n", + " legend=[label for label, counts, state_accuracy in results_backend1_mem],\n", + " title=backend.name(),\n", + " bar_labels=False)" + ] + }, + { + "source": [ + "#### The noisy could be reduced in this experiment by further 4%, however, the orginal results could not be reconstricuted and thus attained or surpassed. At this point, if running this notebook on a shared system (e.g. mybinder.org), remember to log off from your IBM Q account in case you used it for execution on real devices by running the following cell:" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "IBMQ.disable_account()\n", + "IBMQ.delete_account()" + ] + }, + { + "source": [ + "#### Finally, this conclues our experiments conducted with respect to the Mermin Peres Magic Sqaure Game, we elaborated the quantum architecure of the game, presented its mathmatical scheme of operation and quantified the results on real quantum computers. \n", + "\n", + "_Respectfully,\n", + "Dr. Jan-Rainer Lahmann - David Drexlin_" + ], + "cell_type": "markdown", + "metadata": {} } ], "metadata": { @@ -1124,7 +1925,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.5-final" + "version": "3.8.5" } }, "nbformat": 4, From cbbbadcb6c790acd4e9c5620120cfe1e7a4733c5 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Thu, 20 May 2021 23:32:36 +0200 Subject: [PATCH 21/30] fix: order of execution; todo comments --- "Mermin\342\200\223Peres-Game.ipynb" | 505 +++++++++------------------ 1 file changed, 157 insertions(+), 348 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index ee2cd09..16d2b95 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -52,7 +52,7 @@ } ], "source": [ - "import numpy as np\n", + "import numpy as np # comments\n", "import random\n", "import qiskit\n", "\n", @@ -65,6 +65,129 @@ "from qiskit.visualization import plot_histogram " ] }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "#Comment" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [], + "source": [ + "# function that sets up the initial entanglement between Alice and Bob\n", + "def share_bell_state(qc): \n", + " qc.h(0)\n", + " qc.h(1)\n", + " qc.cx(0,2)\n", + " qc.cx(1,3)\n", + " \n", + "# The following functions represent the U(gamma) and V(gamma) controlled Cliffords.\n", + "def U(qc,gamma):\n", + " a = 0\n", + " b = 1\n", + " if gamma==1:\n", + " qc.h(a)\n", + " qc.i(b)\n", + "\n", + " elif gamma==2:\n", + " qc.swap(a,b)\n", + " qc.h(a)\n", + " qc.i(b)\n", + "\n", + " elif gamma==3:\n", + " qc.cx(a,b)\n", + " qc.h(a)\n", + " qc.i(b)\n", + "\n", + "def V(qc,gamma):\n", + " a = 2\n", + " b = 3\n", + " if gamma==1:\n", + " qc.h(a)\n", + " qc.h(b)\n", + "\n", + " if gamma==2:\n", + " qc.swap(a,b)\n", + "\n", + " elif gamma==3:\n", + " qc.z(a)\n", + " qc.z(b)\n", + " qc.cz(a,b)\n", + " qc.h(a)\n", + " qc.h(b)" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "Alice and Bob, are assigned a column or row value of 1, 2, or 3\nAlice assignment (column):\nBob assignment (row):\n3\n" + ] + }, + { + "output_type": "error", + "ename": "NameError", + "evalue": "name 'share_bell_state' is not defined", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 14\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 15\u001b[0m \u001b[0;31m#Generate the Bell state on the circuit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 16\u001b[0;31m \u001b[0mshare_bell_state\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmagicsquare_circuit\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 17\u001b[0m \u001b[0mmagicsquare_circuit\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbarrier\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 18\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mNameError\u001b[0m: name 'share_bell_state' is not defined" + ] + } + ], + "source": [ + "#We will ask the user to select the row and column on Alice and Bob's behalf.\n", + "print(\"Alice and Bob, are assigned a column or row value of 1, 2, or 3\") \n", + "print(\"Alice assignment (column):\")\n", + "alpha = random.randint(1, 4)\n", + "print(\"Bob assignment (row):\")\n", + "beta = random.randint(1, 4)\n", + "print(alpha)\n", + "#Create the quantum register and the classical register to store our final bit values\n", + "q = QuantumRegister(4) # Alice-Qunatum Register (2) and Bob-Qunatum Register (2)\n", + "c = ClassicalRegister(4) \n", + "\n", + "#Create the circuit\n", + "magicsquare_circuit = QuantumCircuit(q,c)\n", + "\n", + "#Generate the Bell state on the circuit\n", + "share_bell_state(magicsquare_circuit)\n", + "magicsquare_circuit.barrier()\n", + "\n", + "#Draw the rest of the circuit based on Alice and Bob's selection \n", + "if 4>alpha>0 and 4>beta>0:\n", + " U(magicsquare_circuit,alpha) \n", + " V(magicsquare_circuit,beta)\n", + " magicsquare_circuit.barrier()\n", + " magicsquare_circuit.measure(q,c)\n", + " \n", + " magicsquare_circuit.draw(output='mpl')\n", + " print(magicsquare_circuit) \n", + "else:\n", + " print(\"please enter values of 1, 2, or 3\")" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "#Comment" + ] + }, { "cell_type": "code", "execution_count": 5, @@ -171,363 +294,24 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": null, "metadata": {}, "outputs": [], "source": [ - "# function to calculate state accuracy\n", - "def get_state_accuracy(counts, expected_states, shots): \n", - " expected_counts = 0\n", - " for state in counts.keys():\n", - " if state in expected_states:\n", - " expected_counts = expected_counts + counts[state]\n", - " state_accuracy = expected_counts / shots\n", - " return str(state_accuracy*100)+\"%\"" + "#Comment" ] }, { "cell_type": "code", - "execution_count": 7, + "execution_count": null, "metadata": {}, "outputs": [], "source": [ - "# function that sets up the initial entanglement between Alice and Bob\n", - "def share_bell_state(qc): \n", - " qc.h(0)\n", - " qc.h(1)\n", - " qc.cx(0,2)\n", - " qc.cx(1,3)\n", - " \n", - "# The following functions represent the U(gamma) and V(gamma) controlled Cliffords.\n", - "def U(qc,gamma):\n", - " a = 0\n", - " b = 1\n", - " if gamma==1:\n", - " qc.h(a)\n", - " qc.i(b)\n", - "\n", - " elif gamma==2:\n", - " qc.swap(a,b)\n", - " qc.h(a)\n", - " qc.i(b)\n", - "\n", - " elif gamma==3:\n", - " qc.cx(a,b)\n", - " qc.h(a)\n", - " qc.i(b)\n", - "\n", - "def V(qc,gamma):\n", - " a = 2\n", - " b = 3\n", - " if gamma==1:\n", - " qc.h(a)\n", - " qc.h(b)\n", - "\n", - " if gamma==2:\n", - " qc.swap(a,b)\n", - "\n", - " elif gamma==3:\n", - " qc.z(a)\n", - " qc.z(b)\n", - " qc.cz(a,b)\n", - " qc.h(a)\n", - " qc.h(b)" - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "metadata": {}, - "outputs": [ - { - "output_type": "stream", - "name": "stdout", - "text": [ - "Alice and Bob, are assigned a column or row value of 1, 2, or 3\n", - "Alice assignment (column):\n", - "Bob assignment (row):\n", - "3\n", - " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", - "q0_0: ┤ H ├──■────────░───■──┤ H ├─░─┤M├─────────\n", - " ├───┤ │ ░ ┌─┴─┐├───┤ ░ └╥┘┌─┐ \n", - "q0_1: ┤ H ├──┼────■───░─┤ X ├┤ I ├─░──╫─┤M├──────\n", - " └───┘┌─┴─┐ │ ░ ├───┤└───┘ ░ ║ └╥┘┌─┐ \n", - "q0_2: ─────┤ X ├──┼───░─┤ H ├──────░──╫──╫─┤M├───\n", - " └───┘┌─┴─┐ ░ ├───┤ ░ ║ ║ └╥┘┌─┐\n", - "q0_3: ──────────┤ X ├─░─┤ H ├──────░──╫──╫──╫─┤M├\n", - " └───┘ ░ └───┘ ░ ║ ║ ║ └╥┘\n", - "c0: 4/════════════════════════════════╩══╩══╩══╩═\n", - " 0 1 2 3 \n", - "{'0010': 1014, '1011': 1009, '0111': 983, '1100': 998, '0000': 1098, '1110': 1020, '1001': 1020, '0101': 1050}\n", - "Resultant quantum circuit solution:\n", - "\n", - "0010\n", - "\n", - "Bring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", - "\n", - "[0 1 0 0]\n", - "\n", - "Compute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\n", - "Compute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n", - "\n", - "(3,)\n", - "Alice's column solution\n", - "[[0]\n", - " [1]\n", - " [0]]\n", - "Winning conditions fullfilled (Odd quantitude of 1)?: True\n", - "\n", - "Bob's row solution\n", - "[0 0 0]\n", - "Winning conditions fullfilled (Even quantitude of 1)?: True\n", - "\n", - "Create matrix to show results\n", - "\n", - "[[ 0. 0. 0.]\n", - " [nan nan 1.]\n", - " [nan nan 0.]]\n", - "\n", - "\n", - "------------------------------------------\n", - "------------------------------------------\n", - "\n", - "\n", - "Quantum circuit solution\n", - "0010\n", - "\n", - "Alice's selection. Column:3\n", - "[[0]\n", - " [1]\n", - " [0]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[0 0 0]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "\n", - "\n", - "[[ 0. 0. 0.]\n", - " [nan nan 1.]\n", - " [nan nan 0.]]\n", - "\n", - "\n", - "------------------------------------------\n", - "\n", - "\n", - "Quantum circuit solution\n", - "1011\n", - "\n", - "Alice's selection. Column:3\n", - "[[1]\n", - " [1]\n", - " [1]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[0 1 1]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "\n", - "\n", - "[[ 0. 1. 1.]\n", - " [nan nan 1.]\n", - " [nan nan 1.]]\n", - "\n", - "\n", - "------------------------------------------\n", - "\n", - "\n", - "Quantum circuit solution\n", - "0111\n", - "\n", - "Alice's selection. Column:3\n", - "[[1]\n", - " [1]\n", - " [1]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[1 0 1]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "\n", - "\n", - "[[ 1. 0. 1.]\n", - " [nan nan 1.]\n", - " [nan nan 1.]]\n", - "\n", - "\n", - "------------------------------------------\n", - "\n", - "\n", - "Quantum circuit solution\n", - "1100\n", - "\n", - "Alice's selection. Column:3\n", - "[[0]\n", - " [0]\n", - " [1]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[1 1 0]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "\n", - "\n", - "[[ 1. 1. 0.]\n", - " [nan nan 0.]\n", - " [nan nan 1.]]\n", - "\n", - "\n", - "------------------------------------------\n", - "\n", - "\n", - "Quantum circuit solution\n", - "0000\n", - "\n", - "Alice's selection. Column:3\n", - "[[0]\n", - " [0]\n", - " [1]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[0 0 0]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "\n", - "\n", - "[[ 0. 0. 0.]\n", - " [nan nan 0.]\n", - " [nan nan 1.]]\n", - "\n", - "\n", - "------------------------------------------\n", - "\n", - "\n", - "Quantum circuit solution\n", - "1110\n", - "\n", - "Alice's selection. Column:3\n", - "[[0]\n", - " [1]\n", - " [0]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[1 1 0]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "\n", - "\n", - "[[ 1. 1. 0.]\n", - " [nan nan 1.]\n", - " [nan nan 0.]]\n", - "\n", - "\n", - "------------------------------------------\n", - "\n", - "\n", - "Quantum circuit solution\n", - "1001\n", - "\n", - "Alice's selection. Column:3\n", - "[[1]\n", - " [0]\n", - " [0]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[0 1 1]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "\n", - "\n", - "[[ 0. 1. 1.]\n", - " [nan nan 0.]\n", - " [nan nan 0.]]\n", - "\n", - "\n", - "------------------------------------------\n", - "\n", - "\n", - "Quantum circuit solution\n", - "0101\n", - "\n", - "Alice's selection. Column:3\n", - "[[1]\n", - " [0]\n", - " [0]]\n", - "\n", - "Bob's circuit solution. Row:1\n", - "[1 0 1]\n", - "\n", - "Victory!\n", - "The following magic square wins the game!\n", - "\n", - "\n", - "[[ 1. 0. 1.]\n", - " [nan nan 0.]\n", - " [nan nan 0.]]\n", - "\n", - "\n", - "------------------------------------------\n", - "\n", - "\n" - ] - }, - { - "output_type": "error", - "ename": "NameError", - "evalue": "name 'get_state_accuracy' is not defined", - "traceback": [ - "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", - "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 31\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresults\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 32\u001b[0m \u001b[0mexpected_states\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcheck_result\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresults\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0malpha\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mbeta\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 33\u001b[0;31m \u001b[0mstate_accuracy\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mget_state_accuracy\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresults\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mexpected_states\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m8192\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 34\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mstate_accuracy\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 35\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;31mNameError\u001b[0m: name 'get_state_accuracy' is not defined" - ] - } - ], - "source": [ - "#We will ask the user to select the row and column on Alice and Bob's behalf.\n", - "print(\"Alice and Bob, are assigned a column or row value of 1, 2, or 3\") \n", - "print(\"Alice assignment (column):\")\n", - "alpha = random.randint(1, 4)\n", - "print(\"Bob assignment (row):\")\n", - "beta = random.randint(1, 4)\n", - "print(alpha)\n", - "#Create the quantum register and the classical register to store our final bit values\n", - "q = QuantumRegister(4) # Alice-Qunatum Register (2) and Bob-Qunatum Register (2)\n", - "c = ClassicalRegister(4) \n", - "\n", - "#Create the circuit\n", - "magicsquare_circuit = QuantumCircuit(q,c)\n", - "\n", - "#Generate the Bell state on the circuit\n", - "share_bell_state(magicsquare_circuit)\n", - "magicsquare_circuit.barrier()\n", - "\n", - "#Draw the rest of the circuit based on Alice and Bob's selection \n", - "if 4>alpha>0 and 4>beta>0:\n", - " U(magicsquare_circuit,alpha) \n", - " V(magicsquare_circuit,beta)\n", - " magicsquare_circuit.barrier()\n", - " magicsquare_circuit.measure(q,c)\n", - " \n", - " magicsquare_circuit.draw(output='mpl')\n", - " print(magicsquare_circuit) \n", - "\n", - " job = execute(magicsquare_circuit, BasicAer.get_backend('qasm_simulator'), optimization_level=3, shots=8192) # run the job simulation\n", - " results = job.result().get_counts()\n", - " print(results)\n", - " expected_states = check_result(results, alpha, beta)\n", - " state_accuracy = get_state_accuracy(results,expected_states,8192)\n", - " print(state_accuracy)\n", - "\n", - "else:\n", - " print(\"please enter values of 1, 2, or 3\")" + "job = execute(magicsquare_circuit, BasicAer.get_backend('qasm_simulator'), optimization_level=3, shots=8192) # run the job simulation\n", + "results = job.result().get_counts()\n", + "print(results)\n", + "expected_states = check_result(results, alpha, beta)\n", + "print(state_accuracy)" ] }, { @@ -887,6 +671,31 @@ "#### To validate these mathematical results, we will run the quantum circuit in one go and compare the respective qubit results" ] }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# function to calculate state accuracy\n", + "def get_state_accuracy(counts, expected_states, shots): \n", + " expected_counts = 0\n", + " for state in counts.keys():\n", + " if state in expected_states:\n", + " expected_counts = expected_counts + counts[state]\n", + " state_accuracy = expected_counts / shots\n", + " return str(state_accuracy*100)+\"%\"" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "#Comment" + ] + }, { "cell_type": "code", "execution_count": 16, From f9d8fe3e78e70c77de1dfdcc8e9afa1a1a403df6 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Sat, 22 May 2021 22:27:36 +0200 Subject: [PATCH 22/30] fix: comments bold --- "Mermin\342\200\223Peres-Game.ipynb" | 183 +++++++++++++-------------- 1 file changed, 91 insertions(+), 92 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 16d2b95..02c73b5 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -9,7 +9,7 @@ }, { "source": [ - "Communication is bilateral. Without proper forms of communication many situations but also games cannot be resolved in the classical sense. TThe Mermin-Peres magic square game is an example of this. It was first proposed by Aravind in 2002, based on the work of Mermin, 1990 and Peres, 1990. It operates, as illustrated below, according to the following scheme: two players, Alice and Bob, play against a game master on a 3x3 grid. They not allowed to communicate. The game master chooses a row and assigns it to Bob and a column is assigned to Alice. \n", + "Communication is bilateral. Without proper forms of communication many situations but also games cannot be resolved in the classical sense. The Mermin-Peres magic square game is an example of this. It was first proposed by Aravind in 2002, based on the work of Mermin, 1990 and Peres, 1990. It operates, as illustrated below, according to the following scheme: two players, Alice and Bob, play against a game master on a 3x3 grid. They not allowed to communicate. The game master chooses a row and assigns it to Bob and a column is assigned to Alice. \n", "\n", "The rules state that the number of tiles placed by Alice in the column must be either zero or two tiles and those placed by Bob in the row must be also either zero or two tiles. However, there must be unity with respect to the intersecting cell, between the row and the column and at least one tile of either player must be placed there. As we can see below, there are various cases when one of the two players places a tile incohrently (i.e. both play a tile in the intersecting cell) and the game is subsequeently lost, so there is no guarantee that the game can be won with a probability of 1. Formally, in a 3x3 grip there are 9 distinct cells, however as presented above the number of total tiles placed is even, as a result Alice and Bob can't agree on a non-overlapping strategy.\n", "\n", @@ -36,47 +36,33 @@ }, { "cell_type": "code", - "execution_count": 38, + "execution_count": 18, "metadata": {}, - "outputs": [ - { - "output_type": "error", - "ename": "ImportError", - "evalue": "cannot import name 'FakeVigo' from 'qiskit' (/Users/david.drexlin@ibm.com/opt/anaconda3/lib/python3.8/site-packages/qiskit/__init__.py)", - "traceback": [ - "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mImportError\u001b[0m Traceback (most recent call last)", - "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 3\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mqiskit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 5\u001b[0;31m \u001b[0;32mfrom\u001b[0m \u001b[0mqiskit\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mQuantumCircuit\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mClassicalRegister\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mQuantumRegister\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexecute\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mBasicAer\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mIBMQ\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mFakeVigo\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 6\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mqiskit\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mproviders\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0maer\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mQasmSimulator\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mUnitarySimulator\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 7\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mqiskit\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mvisualization\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mplot_state_qsphere\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;31mImportError\u001b[0m: cannot import name 'FakeVigo' from 'qiskit' (/Users/david.drexlin@ibm.com/opt/anaconda3/lib/python3.8/site-packages/qiskit/__init__.py)" - ] - } - ], + "outputs": [], "source": [ "import numpy as np # comments\n", - "import random\n", - "import qiskit\n", + "import random # \n", + "import qiskit # import qiskit \n", "\n", - "from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister, execute, BasicAer, IBMQ, FakeVigo\n", + "from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister, execute, BasicAer, IBMQ\n", "from qiskit.providers.aer import QasmSimulator, UnitarySimulator\n", "from qiskit.visualization import plot_state_qsphere\n", - "from qiskit.test.mock import FakeVigo\n", "from qiskit.providers.ibmq import least_busy\n", "from qiskit.ignis.mitigation.measurement import complete_meas_cal,CompleteMeasFitter \n", - "from qiskit.visualization import plot_histogram " + "from qiskit.visualization import plot_histogram \n", + "from qiskit.test.mock import FakeVigo" ] }, { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], "source": [ - "#Comment" - ] + "Starting off, we first bring the circuit in its initial conditions by creating the means to bring the qubits into the bell state. Further, we design the Cliffords U and V used to change the architecture with respect to a gamma, denoting the selected number of row and column for one game instance. " + ], + "cell_type": "markdown", + "metadata": {} }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 5, "metadata": {}, "outputs": [], "source": [ @@ -123,27 +109,38 @@ " qc.h(b)" ] }, + { + "source": [ + "At this point, we can initialise the four required qubits and bring them into the bell state (superposition of the first two qubits and entanglement between the first and third & second and fourth qubit), using share_bell_state function. Subsequently, we let the referee pick the random column and row with is passed to U and V, respectivly. " + ], + "cell_type": "markdown", + "metadata": {} + }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 12, "metadata": {}, "outputs": [ { "output_type": "stream", "name": "stdout", "text": [ - "Alice and Bob, are assigned a column or row value of 1, 2, or 3\nAlice assignment (column):\nBob assignment (row):\n3\n" - ] - }, - { - "output_type": "error", - "ename": "NameError", - "evalue": "name 'share_bell_state' is not defined", - "traceback": [ - "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", - "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 14\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 15\u001b[0m \u001b[0;31m#Generate the Bell state on the circuit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 16\u001b[0;31m \u001b[0mshare_bell_state\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmagicsquare_circuit\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 17\u001b[0m \u001b[0mmagicsquare_circuit\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbarrier\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 18\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;31mNameError\u001b[0m: name 'share_bell_state' is not defined" + "Alice and Bob, are assigned a column or row value of 1, 2, or 3\n", + "Alice assignment (column):\n", + "Bob assignment (row):\n", + "2\n", + " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", + "q9_0: ┤ H ├──■────────░───X──┤ H ├──────░─┤M├─────────\n", + " ├───┤ │ ░ │ ├───┤ ░ └╥┘┌─┐ \n", + "q9_1: ┤ H ├──┼────■───░───X──┤ I ├──────░──╫─┤M├──────\n", + " └───┘┌─┴─┐ │ ░ ┌───┐└───┘┌───┐ ░ ║ └╥┘┌─┐ \n", + "q9_2: ─────┤ X ├──┼───░─┤ Z ├──■──┤ H ├─░──╫──╫─┤M├───\n", + " └───┘┌─┴─┐ ░ ├───┤ │ ├───┤ ░ ║ ║ └╥┘┌─┐\n", + "q9_3: ──────────┤ X ├─░─┤ Z ├──■──┤ H ├─░──╫──╫──╫─┤M├\n", + " └───┘ ░ └───┘ └───┘ ░ ║ ║ ║ └╥┘\n", + "c9: 4/═════════════════════════════════════╩══╩══╩══╩═\n", + " 0 1 2 3 \n", + "haram\n" ] } ], @@ -151,9 +148,9 @@ "#We will ask the user to select the row and column on Alice and Bob's behalf.\n", "print(\"Alice and Bob, are assigned a column or row value of 1, 2, or 3\") \n", "print(\"Alice assignment (column):\")\n", - "alpha = random.randint(1, 4)\n", + "alpha = random.randint(1, 3)\n", "print(\"Bob assignment (row):\")\n", - "beta = random.randint(1, 4)\n", + "beta = random.randint(1, 3)\n", "print(alpha)\n", "#Create the quantum register and the classical register to store our final bit values\n", "q = QuantumRegister(4) # Alice-Qunatum Register (2) and Bob-Qunatum Register (2)\n", @@ -175,22 +172,21 @@ " \n", " magicsquare_circuit.draw(output='mpl')\n", " print(magicsquare_circuit) \n", + " print(\"haram\")\n", "else:\n", " print(\"please enter values of 1, 2, or 3\")" ] }, { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], "source": [ - "#Comment" - ] + "However, at this point we still yet have to execute our circuit, however, prior to this step we must first define functions that present and extend the four qubits to the six game moves played by Alice and BOb. " + ], + "cell_type": "markdown", + "metadata": {} }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 14, "metadata": {}, "outputs": [], "source": [ @@ -250,7 +246,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 15, "metadata": {}, "outputs": [], "source": [ @@ -293,30 +289,36 @@ ] }, { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], "source": [ - "#Comment" - ] + "Consequently, we can execute the circuit created above and directly observe the created results and moves and compare them with moves that solve the Mermin Peres Magic Square Game, we will oberseve that all." + ], + "cell_type": "markdown", + "metadata": {} }, { "cell_type": "code", - "execution_count": null, + "execution_count": 16, "metadata": {}, - "outputs": [], + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "{'0010': 1067, '1100': 1000, '0110': 977, '1000': 1026, '0101': 1001, '0001': 1039, '1011': 1035, '1111': 1047}\nResultant quantum circuit solution:\n\n0010\n\nBring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n\n[0 1 0 0]\n\nCompute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\nCompute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n\n(3,)\nAlice's column solution\n[[0]\n [1]\n [0]]\nWinning conditions fullfilled (Odd quantitude of 1)?: True\n\nBob's row solution\n[0 0 0]\nWinning conditions fullfilled (Even quantitude of 1)?: True\n\nCreate matrix to show results\n\n[[nan 0. nan]\n [nan 1. nan]\n [ 0. 0. 0.]]\n\n\n------------------------------------------\n------------------------------------------\n\n\nQuantum circuit solution\n0010\n\nAlice's selection. Column:2\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:3\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 0. nan]\n [nan 1. nan]\n [ 0. 0. 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1100\n\nAlice's selection. Column:2\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:3\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 0. nan]\n [nan 0. nan]\n [ 1. 1. 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0110\n\nAlice's selection. Column:2\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:3\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 0. nan]\n [nan 1. nan]\n [ 1. 0. 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1000\n\nAlice's selection. Column:2\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:3\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 0. nan]\n [nan 0. nan]\n [ 0. 1. 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0101\n\nAlice's selection. Column:2\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:3\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 1. nan]\n [nan 0. nan]\n [ 1. 0. 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0001\n\nAlice's selection. Column:2\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:3\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 1. nan]\n [nan 0. nan]\n [ 0. 0. 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1011\n\nAlice's selection. Column:2\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:3\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 1. nan]\n [nan 1. nan]\n [ 0. 1. 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1111\n\nAlice's selection. Column:2\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:3\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 1. nan]\n [nan 1. nan]\n [ 1. 1. 0.]]\n\n\n------------------------------------------\n\n\n['0010', '1100', '0110', '1000', '0101', '0001', '1011', '1111']\n" + ] + } + ], "source": [ "job = execute(magicsquare_circuit, BasicAer.get_backend('qasm_simulator'), optimization_level=3, shots=8192) # run the job simulation\n", "results = job.result().get_counts()\n", "print(results)\n", "expected_states = check_result(results, alpha, beta)\n", - "print(state_accuracy)" + "print(expected_states)" ] }, { "source": [ - "#### Indeed, using an ideal quantum simulator, a winning probability of 100% is achieved, which exceeds the highest classical winning probability of 88.99% Beyond that, however, it is desirable to quantify the results on a real IBM quantum computer, which will be realized in the last section." + "Indeed, using an ideal quantum simulator, a winning probability of 100% is achieved, which exceeds the highest classical winning probability of 88.99% Beyond that, however, it is desirable to quantify the results on a real IBM quantum computer, which will be realized in the last section." ], "cell_type": "markdown", "metadata": {} @@ -346,7 +348,7 @@ "\n", "$$ Quantum Solution MPG = Initial Vector * Dynamic Matrix(row, column)$$\n", "\n", - "#### Starting with the first part of the equation, the intial vector, of the MPG circuit is defined by quantum gates as follows:\n" + "Starting with the first part of the equation, the intial vector, of the MPG circuit is defined by quantum gates as follows:\n" ] }, { @@ -379,26 +381,26 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "#### This initial vector is generated, as illustrated above, by the application of two Hadamard gates to the first and second qubits and CNOT gates, with the first and second qubits serving as the control qubit and the third and fourth qubits, respectively, serving as the target qubit. Mathematically, these operations can be described by the following equation: \n", + "This initial vector is generated, as illustrated above, by the application of two Hadamard gates to the first and second qubits and CNOT gates, with the first and second qubits serving as the control qubit and the third and fourth qubits, respectively, serving as the target qubit. Mathematically, these operations can be described by the following equation: \n", "\n", "$$ Initial Vector = (I ⨂ CNOT_{0|2}) ⨂ (CNOT_{1|3} ⨂ I) * (I ⨂ I ⨂ H ⨂ H) * InitialQubits $$\n", "\n", "
\n", "\n", - "#### However, prior to inserting the corresponding matrices into this equation, we introduce the relevant matrices needed for the computation. " + "However, prior to inserting the corresponding matrices into this equation, we introduce the relevant matrices needed for the computation. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "#### The very starting vector state of the MPG circuit, prior to any quantum gate interaction, corresponds to the Dirac bracket |0000> and is hence described by the following vector: \n", + "The very starting vector state of the MPG circuit, prior to any quantum gate interaction, corresponds to the Dirac bracket |0000> and is hence described by the following vector: \n", "$$ InitialQubits= {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}$$" ] }, { "source": [ - "#### In terms of the gates, these are the mathmatical represetations of the relevant gates illustrated above. \n", + "In terms of the gates, these are the mathmatical represetations of the relevant gates illustrated above. \n", "\n", "$$\n", "I = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} \n", @@ -428,7 +430,7 @@ }, { "source": [ - "#### Now we can recall the equation of the initial vectors and insert the matrices:\n", + "Now we can recall the equation of the initial vectors and insert the matrices:\n", "\n", "$$ Initial Vector = (I ⨂ CNOT_{0|2}) ⨂ (CNOT_{1|3} ⨂ I) * (I ⨂ I ⨂ H ⨂ H) * InitialQubits $$\n" ], @@ -485,7 +487,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "#### This resulting vector holds the relevant qubit states that can be further considered and modified in the dynmaic part of the circuit so that valid winning condition can be created, these Qubit states are:\n", + "This resulting vector holds the relevant qubit states that can be further considered and modified in the dynmaic part of the circuit so that valid winning condition can be created, these Qubit states are:\n", "$$= \\frac 1 {2} (|0000\\rangle+|0101\\rangle+|1010\\rangle+|1111\\rangle)$$\n" ] }, @@ -493,7 +495,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "#### Furthermore, when analyzing the initial vector, it becomes apparent that when multiplying any matrix by it, only four relevenate cells per row in the multiplier remain open for consideration, since the others are all multiplied by 0 and thereby likewise drop of with 0% probability as a result. Thus, we can eliminate the majority of the entries whatsoever. As an example we have prepared the matrix below, and denoted all entries with an i, that have to be computed and all others that result in any case with 0 with -.\n", + "Furthermore, when analyzing the initial vector, it becomes apparent that when multiplying any matrix by it, only four relevenate cells per row in the multiplier remain open for consideration, since the others are all multiplied by 0 and thereby likewise drop of with 0% probability as a result. Thus, we can eliminate the majority of the entries whatsoever. As an example we have prepared the matrix below, and denoted all entries with an i, that have to be computed and all others that result in any case with 0 with -.\n", "\n", "$$ = \n", "\\begin{bmatrix}\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\end{bmatrix} \n", @@ -525,14 +527,14 @@ "$$\n", "= \\frac 1 {\\sqrt{8}}\n", "$$\n", - "#### And indeed, this idea is very powerful because it lets us visually see what outcomes may emerge as well asa better idea of which outcomes may fall off and which may be a win in the MPG. As long as postive result emerges the MPG is won, no matter the probability of that play occuring" + "And indeed, this idea is very powerful because it lets us visually see what outcomes may emerge as well asa better idea of which outcomes may fall off and which may be a win in the MPG. As long as postive result emerges the MPG is won, no matter the probability of that play occuring" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "### Now, in order to consider the later, dynamic, part of the equation we will have to select an example. In this case we have selected row: 1 and column: 1. The corresponding quantum circuit is as follows:" + "Now, in order to consider the later, dynamic, part of the equation we will have to select an example. In this case we have selected row: 1 and column: 1. The corresponding quantum circuit is as follows:" ] }, { @@ -567,7 +569,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "#### The translation of these circuit gate operations is in turn equivalent to the following mathematical matrice multiplications:\n", + "The translation of these circuit gate operations is in turn equivalent to the following mathematical matrice multiplications:\n", "$$ DynamicMatrix(1, 1) = H ⨂ H ⨂ I ⨂ H $$\n", "$$\n", "= \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix}\n", @@ -611,7 +613,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "#### Now, as we have both, the inital vector and the dynamic matrix, we can compute the winning plays for Alice and Bob with selected column and row of 1. In prinicple we can apply the rule of the i-rows, that we have established above, to this matrix-vector multiplication, however, in this case for the sake of completness we let all entries remain. But feel free to check the rule!" + "Now, as we have both, the inital vector and the dynamic matrix, we can compute the winning plays for Alice and Bob with selected column and row of 1. In prinicple we can apply the rule of the i-rows, that we have established above, to this matrix-vector multiplication, however, in this case for the sake of completness we let all entries remain. But feel free to check the rule!" ] }, { @@ -664,11 +666,11 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "#### Finally, this vector contains the qubits that can be played to win the game with probability 1, these are:\n", + "Finally, this vector contains the qubits that can be played to win the game with probability 1, these are:\n", "\n", "$$= \\frac 1 {\\sqrt{8}} (|0000\\rangle+|0100\\rangle+|0101\\rangle+|01111\\rangle+|1000\\rangle)-|1011\\rangle)+|1101\\rangle)-|1111\\rangle)$$\n", "\n", - "#### To validate these mathematical results, we will run the quantum circuit in one go and compare the respective qubit results" + "To validate these mathematical results, we will run the quantum circuit in one go and compare the respective qubit results, we can formalize and measure the rate of the winning condition for multiple executions with the state accuracy, which compares the expected, the ideal, states with the resultant states of the circuit and measures deviations: " ] }, { @@ -688,13 +690,11 @@ ] }, { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], "source": [ - "#Comment" - ] + "For row:=1 and column:=1 the following circuit is generated: " + ], + "cell_type": "markdown", + "metadata": {} }, { "cell_type": "code", @@ -751,7 +751,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "#### Evidently, the results of the mathematical calculation match those of the circuit and a win rate of 100% is achieved!" + "Evidently, the results of the mathematical calculation match those of the circuit and a win rate of 100% is achieved!" ] }, { @@ -784,9 +784,9 @@ }, { "source": [ - "#### Thus far all executions of the Quantum circuits were performed on ideal quantum simulators, rather than actual quantum hardware. Real quantum computers have noise, decoherence and other periphery hardware issues, which can reduce the performance and fidelity of the executions. Hence, in order to ascertain the actucal performance and not only the theoretical we will execute our circuit on real quantum computers as well as simulators of real computers that include the above mentioned errors alike. \n", + "Thus far all executions of the Quantum circuits were performed on ideal quantum simulators, rather than actual quantum hardware. Real quantum computers have noise, decoherence and other periphery hardware issues, which can reduce the performance and fidelity of the executions. Hence, in order to ascertain the actucal performance and not only the theoretical we will execute our circuit on real quantum computers as well as simulators of real computers that include the above mentioned errors alike. \n", "\n", - "#### We start off with the non-ideal simulators of the real quantum computers, so called FakeDevices. The use of FakeVigo, simualating the quantum Computer Vigo yields the following results." + "We start off with the non-ideal simulators of the real quantum computers, so called FakeDevices. The use of FakeVigo, simualating the quantum Computer Vigo yields the following results." ], "cell_type": "markdown", "metadata": {} @@ -814,7 +814,7 @@ }, { "source": [ - "#### We can oberserve that simulated noisy devices outperform classical systems and thus indicate that real quantum computers are capable of doing the same. Now in order to evaluate the performance of real devices we select the ibmq_athens computer to benchmark the MPG. Please note that you will have to add your own API token from the IBM Quantum website (https://quantum-computing.ibm.com) so that this cell executes" + "We can oberserve that simulated noisy devices outperform classical systems and thus indicate that real quantum computers are capable of doing the same. Now in order to evaluate the performance of real devices we select the ibmq_athens computer to benchmark the MPG. Please note that you will have to add your own API token from the IBM Quantum website (https://quantum-computing.ibm.com) so that this cell executes" ], "cell_type": "markdown", "metadata": {} @@ -937,7 +937,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "#### As shown above, the MPG circuit outperforms the classical circuit by about 7.8% (88.89% vs. 96.69%) on a real, noisy, quantum computer. Further optimising can be achieved via measurement error mitigation as can be further seen in this [notebook](https://qiskit.org/textbook/ch-quantum-hardware/measurement-error-mitigation.html)." + "As outlined above, the MPG circuit outperforms the classical circuit by about 7.8% (88.89% vs. 96.69%) on a real, noisy, quantum computer. Further optimising can be achieved via measurement error mitigation as can be further seen in this [notebook](https://qiskit.org/textbook/ch-quantum-hardware/measurement-error-mitigation.html)." ] }, { @@ -1693,7 +1693,7 @@ }, { "source": [ - "#### The noisy could be reduced in this experiment by further 4%, however, the orginal results could not be reconstricuted and thus attained or surpassed. At this point, if running this notebook on a shared system (e.g. mybinder.org), remember to log off from your IBM Q account in case you used it for execution on real devices by running the following cell:" + "The noisy could be reduced in this experiment by further 4%, however, the orginal results could not be reconstricuted and thus attained or surpassed. At this point, if running this notebook on a shared system (e.g. mybinder.org). Remember to log off from your IBM Q account in case you used it for execution on real devices by running the following cell:" ], "cell_type": "markdown", "metadata": {} @@ -1710,10 +1710,9 @@ }, { "source": [ - "#### Finally, this conclues our experiments conducted with respect to the Mermin Peres Magic Sqaure Game, we elaborated the quantum architecure of the game, presented its mathmatical scheme of operation and quantified the results on real quantum computers. \n", + "#### Finally, this conclues our experiments with respect to the Mermin Peres Magic Sqaure Game. We elaborated the quantum architecure of the game, presented its scheme of operation and quantified the results on real quantum computers. \n", "\n", - "_Respectfully,\n", - "Dr. Jan-Rainer Lahmann - David Drexlin_" + "_Dr. Jan-Rainer Lahmann - David Drexlin_" ], "cell_type": "markdown", "metadata": {} @@ -1721,8 +1720,8 @@ ], "metadata": { "kernelspec": { - "name": "python385jvsc74a57bd09051d1f6526b9502b8740a4800e72d224ca6abc2bf606f16227017c303a32110", - "display_name": "Python 3.8.5 64-bit ('base': conda)" + "name": "python391jvsc74a57bd0475b625634d962d9a94dcb56ed23109cae51528bdc211180c80ccb0ab7f54442", + "display_name": "Python 3.9.1 64-bit ('quan': conda)" }, "language_info": { "codemirror_mode": { @@ -1734,7 +1733,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.5" + "version": "3.9.1" } }, "nbformat": 4, From 00cbec86adcb720eaafe739ed4390321079068ff Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Tue, 10 Aug 2021 08:51:22 +0200 Subject: [PATCH 23/30] changes --- "Mermin\342\200\223Peres-Game.ipynb" | 17 ++++++++++++----- 1 file changed, 12 insertions(+), 5 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 02c73b5..1679680 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -9,25 +9,32 @@ }, { "source": [ - "Communication is bilateral. Without proper forms of communication many situations but also games cannot be resolved in the classical sense. The Mermin-Peres magic square game is an example of this. It was first proposed by Aravind in 2002, based on the work of Mermin, 1990 and Peres, 1990. It operates, as illustrated below, according to the following scheme: two players, Alice and Bob, play against a game master on a 3x3 grid. They not allowed to communicate. The game master chooses a row and assigns it to Bob and a column is assigned to Alice. \n", + "The ability to read the mind of other humans, know as telepathy, has gripped humanity since the begin of modern times. Flawlessly predicting, however, what other people want to say, feel or will do is largely considered to be pseudosciencte. \n", "\n", - "The rules state that the number of tiles placed by Alice in the column must be either zero or two tiles and those placed by Bob in the row must be also either zero or two tiles. However, there must be unity with respect to the intersecting cell, between the row and the column and at least one tile of either player must be placed there. As we can see below, there are various cases when one of the two players places a tile incohrently (i.e. both play a tile in the intersecting cell) and the game is subsequeently lost, so there is no guarantee that the game can be won with a probability of 1. Formally, in a 3x3 grip there are 9 distinct cells, however as presented above the number of total tiles placed is even, as a result Alice and Bob can't agree on a non-overlapping strategy.\n", + "Interestingly applying the principlies of qunatum entanglment and quantum superposition we can almost fully achieve telepathic abilities. These princiiplies, permit the instantously transfer of quantum information without proper communication between two players in a classiacl sense, even when the players are seperated by many light years where classical communication would take many, many years. \n", "\n", - "However, as an equilibrium strategy Alice and Bob can construct a 3x3 table, containing static examples of moves that can be played by them so that both of them win. However, a conflict between the winning conditions of Alice and Bob is bound to emerge, such that $\\frac 8 {9}$ tiles can be placed without conflict. Consequently, the game has a winning probability of $\\frac 8 {9}$.\n" + "This curucial property can be further utiluzed to solve othwerwise unsolvables-dilemmas in the classiacl sense. This concept can be very well described and illustraed when considerig the Mermin Peres Magic Square Game (MPG) as shown below: \n", + "\n", + "Two players, nicknamed Alice and Bob, play a cooperatice game and are bindingly not allowed to communicate with each other. Each player is assigned a random row and column in a 3x3 grid where they either place zero or two tiles, the position of these tiles is indivually seen irrelavent. However, most curucaiylly there must be unity with respect to the placment of the tiles in the intersecting cell, between the row and the column, only one tile of either player must be placed there. Consequently, with no way to cooridante and commuicate there is no determinsitc guarantee that the game can be won, as we will explore later even with prior collusive behavior its not possible to win with a probability of 1. " ], "cell_type": "markdown", "metadata": {} }, { "source": [ - "

Source: [Twisted Oaks](http://twistedoakstudios.com/blog/Post6536_implementing-quantum-pseudo-telepathy)" + "

Source: [Twisted Oaks](http://twistedoakstudios.com/blog/Post6536_implementing-quantum-pseudo-telepathy) (The game was first proposed by Aravind in 2002, based on the work of Mermin, 1990 and Peres, 1990. It operates,as illustrated above)" ], "cell_type": "markdown", "metadata": {} }, { "source": [ - "However, we can describe the game using a quantum mechanical system that gurantees to solve the game with a probability of 1. We methods such that it seems that Alice and Bob are communucating, hence the name Quantum pseudo-telepathy. This game has been [recently implemented] (https://medium.com/qiskit/this-proof-demonstrates-a-quantum-advantage-even-for-noisy-quantum-computers-b44a738801ad) and our aim is to explain it in detail and extend it to real quantum computers. In the orginal publication, rather than graphical tiles denoted with A & B as in the illustration above, conceptionally equivilant digits +1 and -1 are amployed for comptation. Hereafter, the tile B will be replaced by +1 and tile A by -1. \n", + "More formally, the game operates in a 3x3 grip with 9 distinct cells, however, as presented above the number of total tiles placed is even, as a result Alice and Bob can't agree on a non-overlapping strategy. As outlined, however, as an equilibrium strategy Alice and Bob can construct a 3x3 table, containing static examples of moves that can be played by them so that both of them win. However, a conflict between the winning conditions of Alice and Bob is bound to emerge, such that $\\frac 8 {9}$ tiles can be placed without conflict. Consequently, the game has a winning probability of $\\frac 8 {9}$. \n", + "\n", + "However, we can describe the game using a quantum mechanical system that gurantees to solve the MPG with a probability of 1, quintessitnally introducing mechanisms that seem like telepathy. \n", + "\n", + "\n", + ". This game has been [recently implemented] (https://medium.com/qiskit/this-proof-demonstrates-a-quantum-advantage-even-for-noisy-quantum-computers-b44a738801ad) and our aim is to explain it in detail and extend it to real quantum computers. In the orginal publication, rather than graphical tiles denoted with A & B as in the illustration above, conceptionally equivilant digits +1 and -1 are amployed for comptation. Hereafter, the tile B will be replaced by +1 and tile A by -1. \n", "\n", "To the end of implemenation, the Python Qiskit framework was employed. Subsequently, this notebook will introduce the quantum circuit solving the MPG game, provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game and quantify it's performance on both simulated and real quantum computers. First, for the subsequent circuit definition, evaluation and illustration, we first have to define auxiliary functions and import the corresponding modules from Qiskit and Numpy" ], From 867af5b7aec43873b05b32c78f1ac907ea00c2b1 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Fri, 13 Aug 2021 18:11:34 +0200 Subject: [PATCH 24/30] fix: Friday changes -> TODO Sat & Sun --- "Mermin\342\200\223Peres-Game.ipynb" | 1324 ++++++++++++++++++++++++-- 1 file changed, 1222 insertions(+), 102 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 1679680..87947a3 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -8,38 +8,33 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ - "The ability to read the mind of other humans, know as telepathy, has gripped humanity since the begin of modern times. Flawlessly predicting, however, what other people want to say, feel or will do is largely considered to be pseudosciencte. \n", + "Telepathy, the ability to read the mind of other humans has gripped men since the begining of times. In modern times, however, there are no technolgies and methodologies to read the minds of humans whatsoever and the field is largely conisdered to be pseudoscientific. However, in nature there are at the quantum mechanical level, effects which in an abstracted way mimic and model forms of telepathy. In practice, the application of the quantum mechanical principlies of qunatum entanglment and quantum superposition enable the instantaneous exchange of qunatum information on the collapse of the wave function, without an expplicilt information exchange, this effecr persits even when seperated by many light years where classical communcation would take more than a life time. \n", "\n", - "Interestingly applying the principlies of qunatum entanglment and quantum superposition we can almost fully achieve telepathic abilities. These princiiplies, permit the instantously transfer of quantum information without proper communication between two players in a classiacl sense, even when the players are seperated by many light years where classical communication would take many, many years. \n", + "The application of these principles to problems has developed solutions. which which are considered to be unsolvable or not trivial to solve in non-quantum systems. Within game theory, and for games the with incomplete information this has given players the telepathic ability to derieve information without communcation from the game due to superposition and entanglement, this fact has been dubbed 'quantum pseudo-telepathy'.\n", "\n", - "This curucial property can be further utiluzed to solve othwerwise unsolvables-dilemmas in the classiacl sense. This concept can be very well described and illustraed when considerig the Mermin Peres Magic Square Game (MPG) as shown below: \n", - "\n", - "Two players, nicknamed Alice and Bob, play a cooperatice game and are bindingly not allowed to communicate with each other. Each player is assigned a random row and column in a 3x3 grid where they either place zero or two tiles, the position of these tiles is indivually seen irrelavent. However, most curucaiylly there must be unity with respect to the placment of the tiles in the intersecting cell, between the row and the column, only one tile of either player must be placed there. Consequently, with no way to cooridante and commuicate there is no determinsitc guarantee that the game can be won, as we will explore later even with prior collusive behavior its not possible to win with a probability of 1. " - ], - "cell_type": "markdown", - "metadata": {} + "For further illustration of this \"quantum psuedo-telepathic\" ability let us consider the following problem known under the term the Mermin Peres Magic Square Game (MPG) as visualized below: Two players, nicknamed Alice and Bob, play a cooperative game and are bindingly not allowed to communicate during the game with each other. Both player are assigned a random row and column in a 3x3 grid where they either place zero or two tiles in their respective row / colum. Indivually, they can place their tiles in any or naught of the three possible cells. However, and most curucaiylly there must be unity with respect to the placment of the tiles in the intersecting cell, of the row and the column. Only either one tile of either player may and must be placed there. In consequence and as we will explore further, due to the fact that there is no communication possible, there exist no determinsitc strategy with which the game can be won with a probability of 1, even with a prior collusive agreement to play certain tiles in specific cells, for certain row / column combinations. \n" + ] }, { - "source": [ - "

Source: [Twisted Oaks](http://twistedoakstudios.com/blog/Post6536_implementing-quantum-pseudo-telepathy) (The game was first proposed by Aravind in 2002, based on the work of Mermin, 1990 and Peres, 1990. It operates,as illustrated above)" - ], "cell_type": "markdown", - "metadata": {} + "metadata": {}, + "source": [ + "

Source: [Twisted Oaks](http://twistedoakstudios.com/blog/Post6536_implementing-quantum-pseudo-telepathy) (The game was first proposed by Aravind in 2002, based on the work of Mermin, 1990 and Peres, 1990. Rules and operation as per description above)" + ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ - "More formally, the game operates in a 3x3 grip with 9 distinct cells, however, as presented above the number of total tiles placed is even, as a result Alice and Bob can't agree on a non-overlapping strategy. As outlined, however, as an equilibrium strategy Alice and Bob can construct a 3x3 table, containing static examples of moves that can be played by them so that both of them win. However, a conflict between the winning conditions of Alice and Bob is bound to emerge, such that $\\frac 8 {9}$ tiles can be placed without conflict. Consequently, the game has a winning probability of $\\frac 8 {9}$. \n", - "\n", - "However, we can describe the game using a quantum mechanical system that gurantees to solve the MPG with a probability of 1, quintessitnally introducing mechanisms that seem like telepathy. \n", + "More formally, rather than placing 'graphical' tiles (as seen above), a conceptually equivalent and necessary modifaction for the mathmtical proof of the unsolvable nature of the MPG is the placement of digits of +1 and -1 by Bob and Alice, respectivly instead of 'A' and 'B'. Hereafter, the tile B will be replaced by +1 and tile A by -1, another [recent publication further details on the scheme of operation] (https://medium.com/qiskit/this-proof-demonstrates-a-quantum-advantage-even-for-noisy-quantum-computers-b44a738801ad).\n", "\n", - "\n", - ". This game has been [recently implemented] (https://medium.com/qiskit/this-proof-demonstrates-a-quantum-advantage-even-for-noisy-quantum-computers-b44a738801ad) and our aim is to explain it in detail and extend it to real quantum computers. In the orginal publication, rather than graphical tiles denoted with A & B as in the illustration above, conceptionally equivilant digits +1 and -1 are amployed for comptation. Hereafter, the tile B will be replaced by +1 and tile A by -1. \n", + "Nonetheless, collusive agreements are the classical equilibrium strategy for Alice and Bob. Such collusive agrements can be described as the construction of a $3\\times 3$ table, containing moves that can be played by them. Ultimatly, however, a conflict between the winning conditions of Alice and Bob is bound to emerge in the last cell of the table, so that there is a maximum winning probability of $\\frac 8 {9}$ which corresponds to the number of cells that described in the table.\n", "\n", "To the end of implemenation, the Python Qiskit framework was employed. Subsequently, this notebook will introduce the quantum circuit solving the MPG game, provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game and quantify it's performance on both simulated and real quantum computers. First, for the subsequent circuit definition, evaluation and illustration, we first have to define auxiliary functions and import the corresponding modules from Qiskit and Numpy" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { "cell_type": "code", @@ -61,11 +56,11 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "Starting off, we first bring the circuit in its initial conditions by creating the means to bring the qubits into the bell state. Further, we design the Cliffords U and V used to change the architecture with respect to a gamma, denoting the selected number of row and column for one game instance. " - ], - "cell_type": "markdown", - "metadata": {} + ] }, { "cell_type": "code", @@ -117,11 +112,11 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "At this point, we can initialise the four required qubits and bring them into the bell state (superposition of the first two qubits and entanglement between the first and third & second and fourth qubit), using share_bell_state function. Subsequently, we let the referee pick the random column and row with is passed to U and V, respectivly. " - ], - "cell_type": "markdown", - "metadata": {} + ] }, { "cell_type": "code", @@ -129,8 +124,8 @@ "metadata": {}, "outputs": [ { - "output_type": "stream", "name": "stdout", + "output_type": "stream", "text": [ "Alice and Bob, are assigned a column or row value of 1, 2, or 3\n", "Alice assignment (column):\n", @@ -185,11 +180,11 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "However, at this point we still yet have to execute our circuit, however, prior to this step we must first define functions that present and extend the four qubits to the six game moves played by Alice and BOb. " - ], - "cell_type": "markdown", - "metadata": {} + ] }, { "cell_type": "code", @@ -296,11 +291,11 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "Consequently, we can execute the circuit created above and directly observe the created results and moves and compare them with moves that solve the Mermin Peres Magic Square Game, we will oberseve that all." - ], - "cell_type": "markdown", - "metadata": {} + ] }, { "cell_type": "code", @@ -308,10 +303,228 @@ "metadata": {}, "outputs": [ { - "output_type": "stream", "name": "stdout", + "output_type": "stream", "text": [ - "{'0010': 1067, '1100': 1000, '0110': 977, '1000': 1026, '0101': 1001, '0001': 1039, '1011': 1035, '1111': 1047}\nResultant quantum circuit solution:\n\n0010\n\nBring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n\n[0 1 0 0]\n\nCompute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\nCompute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n\n(3,)\nAlice's column solution\n[[0]\n [1]\n [0]]\nWinning conditions fullfilled (Odd quantitude of 1)?: True\n\nBob's row solution\n[0 0 0]\nWinning conditions fullfilled (Even quantitude of 1)?: True\n\nCreate matrix to show results\n\n[[nan 0. nan]\n [nan 1. nan]\n [ 0. 0. 0.]]\n\n\n------------------------------------------\n------------------------------------------\n\n\nQuantum circuit solution\n0010\n\nAlice's selection. Column:2\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:3\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 0. nan]\n [nan 1. nan]\n [ 0. 0. 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1100\n\nAlice's selection. Column:2\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:3\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 0. nan]\n [nan 0. nan]\n [ 1. 1. 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0110\n\nAlice's selection. Column:2\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:3\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 0. nan]\n [nan 1. nan]\n [ 1. 0. 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1000\n\nAlice's selection. Column:2\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:3\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 0. nan]\n [nan 0. nan]\n [ 0. 1. 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0101\n\nAlice's selection. Column:2\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:3\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 1. nan]\n [nan 0. nan]\n [ 1. 0. 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0001\n\nAlice's selection. Column:2\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:3\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 1. nan]\n [nan 0. nan]\n [ 0. 0. 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1011\n\nAlice's selection. Column:2\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:3\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 1. nan]\n [nan 1. nan]\n [ 0. 1. 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1111\n\nAlice's selection. Column:2\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:3\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[nan 1. nan]\n [nan 1. nan]\n [ 1. 1. 0.]]\n\n\n------------------------------------------\n\n\n['0010', '1100', '0110', '1000', '0101', '0001', '1011', '1111']\n" + "{'0010': 1067, '1100': 1000, '0110': 977, '1000': 1026, '0101': 1001, '0001': 1039, '1011': 1035, '1111': 1047}\n", + "Resultant quantum circuit solution:\n", + "\n", + "0010\n", + "\n", + "Bring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", + "\n", + "[0 1 0 0]\n", + "\n", + "Compute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\n", + "Compute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n", + "\n", + "(3,)\n", + "Alice's column solution\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "Winning conditions fullfilled (Odd quantitude of 1)?: True\n", + "\n", + "Bob's row solution\n", + "[0 0 0]\n", + "Winning conditions fullfilled (Even quantitude of 1)?: True\n", + "\n", + "Create matrix to show results\n", + "\n", + "[[nan 0. nan]\n", + " [nan 1. nan]\n", + " [ 0. 0. 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0010\n", + "\n", + "Alice's selection. Column:2\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[0 0 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan 0. nan]\n", + " [nan 1. nan]\n", + " [ 0. 0. 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1100\n", + "\n", + "Alice's selection. Column:2\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[1 1 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan 0. nan]\n", + " [nan 0. nan]\n", + " [ 1. 1. 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0110\n", + "\n", + "Alice's selection. Column:2\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[1 0 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan 0. nan]\n", + " [nan 1. nan]\n", + " [ 1. 0. 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1000\n", + "\n", + "Alice's selection. Column:2\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[0 1 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan 0. nan]\n", + " [nan 0. nan]\n", + " [ 0. 1. 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0101\n", + "\n", + "Alice's selection. Column:2\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[1 0 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan 1. nan]\n", + " [nan 0. nan]\n", + " [ 1. 0. 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0001\n", + "\n", + "Alice's selection. Column:2\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[0 0 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan 1. nan]\n", + " [nan 0. nan]\n", + " [ 0. 0. 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1011\n", + "\n", + "Alice's selection. Column:2\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[0 1 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan 1. nan]\n", + " [nan 1. nan]\n", + " [ 0. 1. 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1111\n", + "\n", + "Alice's selection. Column:2\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:3\n", + "[1 1 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[nan 1. nan]\n", + " [nan 1. nan]\n", + " [ 1. 1. 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "['0010', '1100', '0110', '1000', '0101', '0001', '1011', '1111']\n" ] } ], @@ -324,11 +537,11 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "Indeed, using an ideal quantum simulator, a winning probability of 100% is achieved, which exceeds the highest classical winning probability of 88.99% Beyond that, however, it is desirable to quantify the results on a real IBM quantum computer, which will be realized in the last section." - ], - "cell_type": "markdown", - "metadata": {} + ] }, { "cell_type": "markdown", @@ -364,16 +577,16 @@ "metadata": {}, "outputs": [ { - "output_type": "execute_result", "data": { + "image/png": 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\n" + ] }, + "execution_count": 11, "metadata": {}, - "execution_count": 11 + "output_type": "execute_result" } ], "source": [ @@ -406,6 +619,8 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "In terms of the gates, these are the mathmatical represetations of the relevant gates illustrated above. \n", "\n", @@ -431,18 +646,16 @@ "
\n", "The \"i\" in CNOT(i, i+2) refers to the control qubit in the first instance and to the target qubit in the second instance under the condition that there are at least two qubits in between such as $CNOT_(1,3)$\n", "\n" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "Now we can recall the equation of the initial vectors and insert the matrices:\n", "\n", "$$ Initial Vector = (I ⨂ CNOT_{0|2}) ⨂ (CNOT_{1|3} ⨂ I) * (I ⨂ I ⨂ H ⨂ H) * InitialQubits $$\n" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { "cell_type": "markdown", @@ -550,16 +763,16 @@ "metadata": {}, "outputs": [ { - "output_type": "execute_result", "data": { + "image/png": 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\n" + ] }, + "execution_count": 14, "metadata": {}, - "execution_count": 14 + "output_type": "execute_result" } ], "source": [ @@ -697,11 +910,11 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "For row:=1 and column:=1 the following circuit is generated: " - ], - "cell_type": "markdown", - "metadata": {} + ] }, { "cell_type": "code", @@ -709,16 +922,16 @@ "metadata": {}, "outputs": [ { - "output_type": "execute_result", "data": { + "image/png": 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", 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\n" + ] }, + "execution_count": 16, "metadata": {}, - "execution_count": 16 + "output_type": "execute_result" } ], "source": [ @@ -739,17 +952,234 @@ "metadata": {}, "outputs": [ { - "output_type": "stream", "name": "stdout", + "output_type": "stream", "text": [ - "Resultant quantum circuit solution:\n\n0000\n\nBring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n\n[0 0 0 0]\n\nCompute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\nCompute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n\n(3,)\nAlice's column solution\n[[0]\n [0]\n [1]]\nWinning conditions fullfilled (Odd quantitude of 1)?: True\n\nBob's row solution\n[0 0 0]\nWinning conditions fullfilled (Even quantitude of 1)?: True\n\nCreate matrix to show results\n\n[[ 0. 0. 0.]\n [ 0. nan nan]\n [ 1. nan nan]]\n\n\n------------------------------------------\n------------------------------------------\n\n\nQuantum circuit solution\n0000\n\nAlice's selection. Column:1\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 0. 0.]\n [ 0. nan nan]\n [ 1. nan nan]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1111\n\nAlice's selection. Column:1\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 1. 0.]\n [ 1. nan nan]\n [ 1. nan nan]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0101\n\nAlice's selection. Column:1\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 0. 1.]\n [ 0. nan nan]\n [ 0. nan nan]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1000\n\nAlice's selection. Column:1\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 1. 1.]\n [ 0. nan nan]\n [ 1. nan nan]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0010\n\nAlice's selection. Column:1\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 0. 0.]\n [ 1. nan nan]\n [ 0. nan nan]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0111\n\nAlice's selection. Column:1\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 0. 1.]\n [ 1. nan nan]\n [ 1. nan nan]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1010\n\nAlice's selection. Column:1\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 1. 1.]\n [ 1. nan nan]\n [ 0. nan nan]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1101\n\nAlice's selection. Column:1\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 1. 0.]\n [ 0. nan nan]\n [ 0. nan nan]]\n\n\n------------------------------------------\n\n\nAccuracy: 100.0%\n" - ] - } - ], - "source": [ - "backend = BasicAer.get_backend('qasm_simulator')\n", - "results = execute(qc, backend, optimization_level=3, shots=8192).result().get_counts()\n", - "expected_states = check_result(results, 1, 1)\n", + "Resultant quantum circuit solution:\n", + "\n", + "0000\n", + "\n", + "Bring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", + "\n", + "[0 0 0 0]\n", + "\n", + "Compute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\n", + "Compute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n", + "\n", + "(3,)\n", + "Alice's column solution\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "Winning conditions fullfilled (Odd quantitude of 1)?: True\n", + "\n", + "Bob's row solution\n", + "[0 0 0]\n", + "Winning conditions fullfilled (Even quantitude of 1)?: True\n", + "\n", + "Create matrix to show results\n", + "\n", + "[[ 0. 0. 0.]\n", + " [ 0. nan nan]\n", + " [ 1. nan nan]]\n", + "\n", + "\n", + "------------------------------------------\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0000\n", + "\n", + "Alice's selection. Column:1\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 0. 0.]\n", + " [ 0. nan nan]\n", + " [ 1. nan nan]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1111\n", + "\n", + "Alice's selection. Column:1\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 1. 0.]\n", + " [ 1. nan nan]\n", + " [ 1. nan nan]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0101\n", + "\n", + "Alice's selection. Column:1\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 0. 1.]\n", + " [ 0. nan nan]\n", + " [ 0. nan nan]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1000\n", + "\n", + "Alice's selection. Column:1\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 1. 1.]\n", + " [ 0. nan nan]\n", + " [ 1. nan nan]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0010\n", + "\n", + "Alice's selection. Column:1\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 0. 0.]\n", + " [ 1. nan nan]\n", + " [ 0. nan nan]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0111\n", + "\n", + "Alice's selection. Column:1\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 0. 1.]\n", + " [ 1. nan nan]\n", + " [ 1. nan nan]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1010\n", + "\n", + "Alice's selection. Column:1\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 1. 1.]\n", + " [ 1. nan nan]\n", + " [ 0. nan nan]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1101\n", + "\n", + "Alice's selection. Column:1\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 1. 0.]\n", + " [ 0. nan nan]\n", + " [ 0. nan nan]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Accuracy: 100.0%\n" + ] + } + ], + "source": [ + "backend = BasicAer.get_backend('qasm_simulator')\n", + "results = execute(qc, backend, optimization_level=3, shots=8192).result().get_counts()\n", + "expected_states = check_result(results, 1, 1)\n", "state_accuracy = get_state_accuracy(results,expected_states,8192)\n", "print(\"Accuracy: \"+state_accuracy)" ] @@ -762,6 +1192,8 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "------------------------------------------------------------------------------------------------------------------------------------------------------------------\n", "To reproduce these operations for other configurations, it is recommended to use the Unitary simulator - follow this simple guide:\n", @@ -778,9 +1210,7 @@ "result = execute(qc, BasicAer.get_backend('unitary_simulator')).result().get_unitary(qc)
\n", "array_to_latex(result, pretext=\"\\\\text{Circuit = }\\n\") \n", "------------------------------------------------------------------------------------------------------------------------------------------------------------------" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { "cell_type": "markdown", @@ -790,13 +1220,13 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "Thus far all executions of the Quantum circuits were performed on ideal quantum simulators, rather than actual quantum hardware. Real quantum computers have noise, decoherence and other periphery hardware issues, which can reduce the performance and fidelity of the executions. Hence, in order to ascertain the actucal performance and not only the theoretical we will execute our circuit on real quantum computers as well as simulators of real computers that include the above mentioned errors alike. \n", "\n", "We start off with the non-ideal simulators of the real quantum computers, so called FakeDevices. The use of FakeVigo, simualating the quantum Computer Vigo yields the following results." - ], - "cell_type": "markdown", - "metadata": {} + ] }, { "cell_type": "code", @@ -804,10 +1234,355 @@ "metadata": {}, "outputs": [ { - "output_type": "stream", "name": "stdout", + "output_type": "stream", "text": [ - "Resultant quantum circuit solution:\n\n0000\n\nBring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n\n[0 0 0 0]\n\nCompute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\nCompute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n\n(3,)\nAlice's column solution\n[[0]\n [0]\n [1]]\nWinning conditions fullfilled (Odd quantitude of 1)?: True\n\nBob's row solution\n[0 0 0]\nWinning conditions fullfilled (Even quantitude of 1)?: True\n\nCreate matrix to show results\n\n[[ 0. 0. 0.]\n [nan nan 0.]\n [nan nan 1.]]\n\n\n------------------------------------------\n------------------------------------------\n\n\nQuantum circuit solution\n0000\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 0. 0.]\n [nan nan 0.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0001\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0010\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 0. 0.]\n [nan nan 1.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0011\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0100\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0101\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 0. 1.]\n [nan nan 0.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0110\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0111\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 0. 1.]\n [nan nan 1.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1000\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1001\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 1. 1.]\n [nan nan 0.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1010\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1011\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 1. 1.]\n [nan nan 1.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1100\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 1. 0.]\n [nan nan 0.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1101\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1110\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 1. 0.]\n [nan nan 1.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1111\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\n\n\n------------------------------------------\n\n\n93.00537109375%\n" + "Resultant quantum circuit solution:\n", + "\n", + "0000\n", + "\n", + "Bring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", + "\n", + "[0 0 0 0]\n", + "\n", + "Compute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\n", + "Compute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n", + "\n", + "(3,)\n", + "Alice's column solution\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "Winning conditions fullfilled (Odd quantitude of 1)?: True\n", + "\n", + "Bob's row solution\n", + "[0 0 0]\n", + "Winning conditions fullfilled (Even quantitude of 1)?: True\n", + "\n", + "Create matrix to show results\n", + "\n", + "[[ 0. 0. 0.]\n", + " [nan nan 0.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0000\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 0. 0.]\n", + " [nan nan 0.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0001\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0010\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 0. 0.]\n", + " [nan nan 1.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0011\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0100\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0101\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 0. 1.]\n", + " [nan nan 0.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0110\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0111\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 0. 1.]\n", + " [nan nan 1.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1000\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1001\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 1. 1.]\n", + " [nan nan 0.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1010\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1011\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 1. 1.]\n", + " [nan nan 1.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1100\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 1. 0.]\n", + " [nan nan 0.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1101\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1110\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 1. 0.]\n", + " [nan nan 1.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1111\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "93.00537109375%\n" ] } ], @@ -820,11 +1595,11 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "We can oberserve that simulated noisy devices outperform classical systems and thus indicate that real quantum computers are capable of doing the same. Now in order to evaluate the performance of real devices we select the ibmq_athens computer to benchmark the MPG. Please note that you will have to add your own API token from the IBM Quantum website (https://quantum-computing.ibm.com) so that this cell executes" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { "cell_type": "code", @@ -832,8 +1607,8 @@ "metadata": {}, "outputs": [ { - "output_type": "stream", "name": "stderr", + "output_type": "stream", "text": [ "configrc.store_credentials:WARNING:2021-05-19 14:15:02,195: Credentials already present. Set overwrite=True to overwrite.\n" ] @@ -849,10 +1624,11 @@ "metadata": {}, "outputs": [ { - "output_type": "stream", "name": "stderr", + "output_type": "stream", "text": [ - "/Users/david.drexlin@ibm.com/opt/anaconda3/lib/python3.8/site-packages/qiskit/providers/ibmq/ibmqfactory.py:192: UserWarning: Timestamps in IBMQ backend properties, jobs, and job results are all now in local time instead of UTC.\n warnings.warn('Timestamps in IBMQ backend properties, jobs, and job results '\n" + "/Users/david.drexlin@ibm.com/opt/anaconda3/lib/python3.8/site-packages/qiskit/providers/ibmq/ibmqfactory.py:192: UserWarning: Timestamps in IBMQ backend properties, jobs, and job results are all now in local time instead of UTC.\n", + " warnings.warn('Timestamps in IBMQ backend properties, jobs, and job results '\n" ] } ], @@ -868,10 +1644,354 @@ "metadata": {}, "outputs": [ { - "output_type": "stream", "name": "stdout", + "output_type": "stream", "text": [ - "Resultant quantum circuit solution:\n\n0000\n\nBring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n\n[0 0 0 0]\n\nCompute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\nCompute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n\n(3,)\nAlice's column solution\n[[0]\n [0]\n [1]]\nWinning conditions fullfilled (Odd quantitude of 1)?: True\n\nBob's row solution\n[0 0 0]\nWinning conditions fullfilled (Even quantitude of 1)?: True\n\nCreate matrix to show results\n\n[[ 0. 0. 0.]\n [nan nan 0.]\n [nan nan 1.]]\n\n\n------------------------------------------\n------------------------------------------\n\n\nQuantum circuit solution\n0000\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 0. 0.]\n [nan nan 0.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0001\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0010\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 0. 0.]\n [nan nan 1.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0011\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[0 0 0]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0100\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0101\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 0. 1.]\n [nan nan 0.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0110\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n0111\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[1 0 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 0. 1.]\n [nan nan 1.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1000\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1001\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 1. 1.]\n [nan nan 0.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1010\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1011\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[0 1 1]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 0. 1. 1.]\n [nan nan 1.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1100\n\nAlice's selection. Column:3\n[[0]\n [0]\n [1]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 1. 0.]\n [nan nan 0.]\n [nan nan 1.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1101\n\nAlice's selection. Column:3\n[[1]\n [0]\n [0]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1110\n\nAlice's selection. Column:3\n[[0]\n [1]\n [0]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\nVictory!\nThe following magic square wins the game!\n\n\n[[ 1. 1. 0.]\n [nan nan 1.]\n [nan nan 0.]]\n\n\n------------------------------------------\n\n\nQuantum circuit solution\n1111\n\nAlice's selection. Column:3\n[[1]\n [1]\n [1]]\n\nBob's circuit solution. Row:1\n[1 1 0]\n\n\n\n------------------------------------------\n\n\n" + "Resultant quantum circuit solution:\n", + "\n", + "0000\n", + "\n", + "Bring array in correct bit order i.e. from incoming y2,y1,x2,x1 -> x1,x2,y1,y2\n", + "\n", + "[0 0 0 0]\n", + "\n", + "Compute missing x3 with x1+x2+1 mod 2 for Alice and save to array for further processing\n", + "Compute missing y3 with y1+y2 mod 2 for Bob and save to array for further processing\n", + "\n", + "(3,)\n", + "Alice's column solution\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "Winning conditions fullfilled (Odd quantitude of 1)?: True\n", + "\n", + "Bob's row solution\n", + "[0 0 0]\n", + "Winning conditions fullfilled (Even quantitude of 1)?: True\n", + "\n", + "Create matrix to show results\n", + "\n", + "[[ 0. 0. 0.]\n", + " [nan nan 0.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0000\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 0. 0.]\n", + " [nan nan 0.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0001\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0010\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 0. 0.]\n", + " [nan nan 1.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0011\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 0 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0100\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0101\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 0. 1.]\n", + " [nan nan 0.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0110\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "0111\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 0 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 0. 1.]\n", + " [nan nan 1.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1000\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1001\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 1. 1.]\n", + " [nan nan 0.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1010\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1011\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[0 1 1]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 0. 1. 1.]\n", + " [nan nan 1.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1100\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [0]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 1. 0.]\n", + " [nan nan 0.]\n", + " [nan nan 1.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1101\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [0]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1110\n", + "\n", + "Alice's selection. Column:3\n", + "[[0]\n", + " [1]\n", + " [0]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "Victory!\n", + "The following magic square wins the game!\n", + "\n", + "\n", + "[[ 1. 1. 0.]\n", + " [nan nan 1.]\n", + " [nan nan 0.]]\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n", + "Quantum circuit solution\n", + "1111\n", + "\n", + "Alice's selection. Column:3\n", + "[[1]\n", + " [1]\n", + " [1]]\n", + "\n", + "Bob's circuit solution. Row:1\n", + "[1 1 0]\n", + "\n", + "\n", + "\n", + "------------------------------------------\n", + "\n", + "\n" ] } ], @@ -888,8 +2008,8 @@ "metadata": {}, "outputs": [ { - "output_type": "stream", "name": "stdout", + "output_type": "stream", "text": [ "68.51806640625%\n" ] @@ -953,8 +2073,8 @@ "metadata": {}, "outputs": [ { - "output_type": "stream", "name": "stdout", + "output_type": "stream", "text": [ "Resultant quantum circuit solution:\n", "\n", @@ -1652,16 +2772,16 @@ ] }, { - "output_type": "execute_result", "data": { + "image/png": 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", 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\n" + ] }, + "execution_count": 32, "metadata": {}, - "execution_count": 32 + "output_type": "execute_result" } ], "source": [ @@ -1699,11 +2819,11 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "The noisy could be reduced in this experiment by further 4%, however, the orginal results could not be reconstricuted and thus attained or surpassed. At this point, if running this notebook on a shared system (e.g. mybinder.org). Remember to log off from your IBM Q account in case you used it for execution on real devices by running the following cell:" - ], - "cell_type": "markdown", - "metadata": {} + ] }, { "cell_type": "code", @@ -1716,19 +2836,19 @@ ] }, { + "cell_type": "markdown", + "metadata": {}, "source": [ "#### Finally, this conclues our experiments with respect to the Mermin Peres Magic Sqaure Game. We elaborated the quantum architecure of the game, presented its scheme of operation and quantified the results on real quantum computers. \n", "\n", "_Dr. Jan-Rainer Lahmann - David Drexlin_" - ], - "cell_type": "markdown", - "metadata": {} + ] } ], "metadata": { "kernelspec": { - "name": "python391jvsc74a57bd0475b625634d962d9a94dcb56ed23109cae51528bdc211180c80ccb0ab7f54442", - "display_name": "Python 3.9.1 64-bit ('quan': conda)" + "display_name": "Python 3.9.1 64-bit ('quan': conda)", + "name": "python391jvsc74a57bd0475b625634d962d9a94dcb56ed23109cae51528bdc211180c80ccb0ab7f54442" }, "language_info": { "codemirror_mode": { From 611daf264aa4be6d630e3b273a8628d68860acff Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Sun, 15 Aug 2021 10:17:58 +0200 Subject: [PATCH 25/30] fix: math --- "Mermin\342\200\223Peres-Game.ipynb" | 18 ++++++++++++++++-- 1 file changed, 16 insertions(+), 2 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 87947a3..32567b8 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -29,9 +29,23 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "More formally, rather than placing 'graphical' tiles (as seen above), a conceptually equivalent and necessary modifaction for the mathmtical proof of the unsolvable nature of the MPG is the placement of digits of +1 and -1 by Bob and Alice, respectivly instead of 'A' and 'B'. Hereafter, the tile B will be replaced by +1 and tile A by -1, another [recent publication further details on the scheme of operation] (https://medium.com/qiskit/this-proof-demonstrates-a-quantum-advantage-even-for-noisy-quantum-computers-b44a738801ad).\n", + "More formally, rather than placing 'graphical' tiles (as seen above), a conceptually equivalent and necessary modifaction for the mathmtical proof of the unsolvable nature of the MPG is the placement of digits of +1 and -1 by Bob and Alice, respectivly instead of 'A' and 'B' as well as. Hereafter, the tile B will be replaced by +1 and tile A by -1, another [recent publication further details on the scheme of operation] (https://medium.com/qiskit/this-proof-demonstrates-a-quantum-advantage-even-for-noisy-quantum-computers-b44a738801ad).\n", "\n", - "Nonetheless, collusive agreements are the classical equilibrium strategy for Alice and Bob. Such collusive agrements can be described as the construction of a $3\\times 3$ table, containing moves that can be played by them. Ultimatly, however, a conflict between the winning conditions of Alice and Bob is bound to emerge in the last cell of the table, so that there is a maximum winning probability of $\\frac 8 {9}$ which corresponds to the number of cells that described in the table.\n", + "Mathmatically, the rules that either zero or two tiles must be placed can be mathmatically translated into the fact \n", + "\n", + " game rules can be stated \n", + "\n", + "we now formualte three conditions which must be met, so that the game is won:\n", + "\n", + "1. $\\overset{3}{\\underset{i = 1}{\\Pi}} m_{i,j} = -1 \\quad \\forall j \\in{1,2,3} $\n", + "2. $\\overset{3}{\\underset{j = 1}{\\Pi}} m_{i,j} = +1 \\quad \\forall i \\in{1,2,3} $\n", + "\n", + "3. For the third rule, and the fact of the intersection between the column and row a confict is bound to emerge as the $m_{i,j}$ variables are commuting and thus can't describe both winning conditions:\n", + "- $\\overset{3}{\\underset{j = 1}{\\Pi}}\\overset{3}{\\underset{i = 1}{\\Pi}} m_{i,j} = -1$\n", + "- $\\overset{3}{\\underset{i = 1}{\\Pi}}\\overset{3}{\\underset{j = 1}{\\Pi}} m_{i,j} = +1$ \n", + "- $-1 = \\overset{3}{\\underset{i,j = 1}{\\Pi}}m_{i,j} = +1 $\n", + "\n", + "Nevertheless, a partial table of collusive agreements can be constructed with only one out of nine cells conflicting, adhering to such a table they can classically achieve a winning rate of $\\frac{8}{9}$ However, when applying quantum mechanical principles, we will show and obtain an unambiguous winning strategy for Alice and Bob, indicating that something classically \\emph{impossible} is happening\n", "\n", "To the end of implemenation, the Python Qiskit framework was employed. Subsequently, this notebook will introduce the quantum circuit solving the MPG game, provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game and quantify it's performance on both simulated and real quantum computers. First, for the subsequent circuit definition, evaluation and illustration, we first have to define auxiliary functions and import the corresponding modules from Qiskit and Numpy" ] From 61114a02cb7c496803919ff8b4659370ff5de70d Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Sun, 15 Aug 2021 16:57:12 +0200 Subject: [PATCH 26/30] fix: text. todo typo --- "Mermin\342\200\223Peres-Game.ipynb" | 28 ++++++++++++++++++++-------- 1 file changed, 20 insertions(+), 8 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 32567b8..15b8aec 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -45,9 +45,14 @@ "- $\\overset{3}{\\underset{i = 1}{\\Pi}}\\overset{3}{\\underset{j = 1}{\\Pi}} m_{i,j} = +1$ \n", "- $-1 = \\overset{3}{\\underset{i,j = 1}{\\Pi}}m_{i,j} = +1 $\n", "\n", - "Nevertheless, a partial table of collusive agreements can be constructed with only one out of nine cells conflicting, adhering to such a table they can classically achieve a winning rate of $\\frac{8}{9}$ However, when applying quantum mechanical principles, we will show and obtain an unambiguous winning strategy for Alice and Bob, indicating that something classically \\emph{impossible} is happening\n", - "\n", - "To the end of implemenation, the Python Qiskit framework was employed. Subsequently, this notebook will introduce the quantum circuit solving the MPG game, provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game and quantify it's performance on both simulated and real quantum computers. First, for the subsequent circuit definition, evaluation and illustration, we first have to define auxiliary functions and import the corresponding modules from Qiskit and Numpy" + "Nevertheless, a partial table of collusive agreements can be constructed with only one out of nine cells conflicting, adhering to such a table they can classically achieve a winning rate of $\\frac{8}{9}$ However, when applying quantum mechanical principles, we will show and obtain an unambiguous winning strategy for Alice and Bob, indicating that something classically impossible is happening." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "To the end of implemenation, the Python Qiskit framework was employed. Subsequently, this notebook will introduce the quantum circuit solving the MPG game, provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game and quantify it's performance on both simulated and real quantum computers. First, for the subsequent circuit definition, evaluation and illustration, we first have to define auxiliary functions and import the corresponding modules from Qiskit and Numpy." ] }, { @@ -73,7 +78,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Starting off, we first bring the circuit in its initial conditions by creating the means to bring the qubits into the bell state. Further, we design the Cliffords U and V used to change the architecture with respect to a gamma, denoting the selected number of row and column for one game instance. " + " In general we employ four qubits to model the MPG. Two qubits are assigned to Alice and two qubits to Bob, while the state and move of the third qubit can be logically derieved from the first two. In a first step, we prepare code in order that these four qubits are brought into a state of superposition and entanglement. Further, we design the Cliffords U and V which conditionally change the architecture of the circuit and concomittant different rotations on the axis of the qubit with respect to a gamma, representing the randomily assigned row / column." ] }, { @@ -129,7 +134,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "At this point, we can initialise the four required qubits and bring them into the bell state (superposition of the first two qubits and entanglement between the first and third & second and fourth qubit), using share_bell_state function. Subsequently, we let the referee pick the random column and row with is passed to U and V, respectivly. " + "At this point, we can initialise the four qubits and bring them into the bell state (superposition of the first two qubits and entanglement between the first and third & second and fourth qubit), using the prior created share_bell_state function. Subsequently, we let the referee pick the random column and row with is passed to U and V, respectivly and visualize the as-is circuit. " ] }, { @@ -197,7 +202,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "However, at this point we still yet have to execute our circuit, however, prior to this step we must first define functions that present and extend the four qubits to the six game moves played by Alice and BOb. " + "However, at this point we still yet have to execute our circuit, however, prior to this step we must first define functions that extent the four qubits of Alice and Bob to the six game moves played and implement measures to determine the win / lose outcome of the played game. " ] }, { @@ -260,6 +265,13 @@ " return expected_states" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Moreoever, for the purposes of explainability we describe and present the winning combinations for the selected row / column and from the four-qubit tangiable output the third qubits are computed. " + ] + }, { "cell_type": "code", "execution_count": 15, @@ -308,7 +320,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Consequently, we can execute the circuit created above and directly observe the created results and moves and compare them with moves that solve the Mermin Peres Magic Square Game, we will oberseve that all." + "Consequently, we can execute the circuit created above and directly observe the created results and moves and compare them with moves that solve the Mermin Peres Magic Square Game, we will oberseve that all of the possible moves that, both, Alice and Bob made fulfill the winning conditions of the MPE." ] }, { @@ -2853,7 +2865,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "#### Finally, this conclues our experiments with respect to the Mermin Peres Magic Sqaure Game. We elaborated the quantum architecure of the game, presented its scheme of operation and quantified the results on real quantum computers. \n", + "#### Finally, this conclues our experiments on the Mermin Peres Magic Sqaure Game. We elaborated the quantum architecure of the game, presented its scheme of operation and quantified the results on real quantum computers. \n", "\n", "_Dr. Jan-Rainer Lahmann - David Drexlin_" ] From 53301b45779fa4cdaa578f142e96feee69c948b7 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Wed, 18 Aug 2021 09:07:01 +0200 Subject: [PATCH 27/30] print --- "Mermin\342\200\223Peres-Game.ipynb" | 521 +++++++++++++-------------- 1 file changed, 260 insertions(+), 261 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 15b8aec..1a5509b 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -2,32 +2,31 @@ "cells": [ { "cell_type": "markdown", - "metadata": {}, "source": [ "# Quantum pseudo-telepathy: Mermin Peres Magic Square Game" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "Telepathy, the ability to read the mind of other humans has gripped men since the begining of times. In modern times, however, there are no technolgies and methodologies to read the minds of humans whatsoever and the field is largely conisdered to be pseudoscientific. However, in nature there are at the quantum mechanical level, effects which in an abstracted way mimic and model forms of telepathy. In practice, the application of the quantum mechanical principlies of qunatum entanglment and quantum superposition enable the instantaneous exchange of qunatum information on the collapse of the wave function, without an expplicilt information exchange, this effecr persits even when seperated by many light years where classical communcation would take more than a life time. \n", "\n", "The application of these principles to problems has developed solutions. which which are considered to be unsolvable or not trivial to solve in non-quantum systems. Within game theory, and for games the with incomplete information this has given players the telepathic ability to derieve information without communcation from the game due to superposition and entanglement, this fact has been dubbed 'quantum pseudo-telepathy'.\n", "\n", "For further illustration of this \"quantum psuedo-telepathic\" ability let us consider the following problem known under the term the Mermin Peres Magic Square Game (MPG) as visualized below: Two players, nicknamed Alice and Bob, play a cooperative game and are bindingly not allowed to communicate during the game with each other. Both player are assigned a random row and column in a 3x3 grid where they either place zero or two tiles in their respective row / colum. Indivually, they can place their tiles in any or naught of the three possible cells. However, and most curucaiylly there must be unity with respect to the placment of the tiles in the intersecting cell, of the row and the column. Only either one tile of either player may and must be placed there. In consequence and as we will explore further, due to the fact that there is no communication possible, there exist no determinsitc strategy with which the game can be won with a probability of 1, even with a prior collusive agreement to play certain tiles in specific cells, for certain row / column combinations. \n" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "

Source: [Twisted Oaks](http://twistedoakstudios.com/blog/Post6536_implementing-quantum-pseudo-telepathy) (The game was first proposed by Aravind in 2002, based on the work of Mermin, 1990 and Peres, 1990. Rules and operation as per description above)" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "More formally, rather than placing 'graphical' tiles (as seen above), a conceptually equivalent and necessary modifaction for the mathmtical proof of the unsolvable nature of the MPG is the placement of digits of +1 and -1 by Bob and Alice, respectivly instead of 'A' and 'B' as well as. Hereafter, the tile B will be replaced by +1 and tile A by -1, another [recent publication further details on the scheme of operation] (https://medium.com/qiskit/this-proof-demonstrates-a-quantum-advantage-even-for-noisy-quantum-computers-b44a738801ad).\n", "\n", @@ -46,20 +45,19 @@ "- $-1 = \\overset{3}{\\underset{i,j = 1}{\\Pi}}m_{i,j} = +1 $\n", "\n", "Nevertheless, a partial table of collusive agreements can be constructed with only one out of nine cells conflicting, adhering to such a table they can classically achieve a winning rate of $\\frac{8}{9}$ However, when applying quantum mechanical principles, we will show and obtain an unambiguous winning strategy for Alice and Bob, indicating that something classically impossible is happening." - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "To the end of implemenation, the Python Qiskit framework was employed. Subsequently, this notebook will introduce the quantum circuit solving the MPG game, provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game and quantify it's performance on both simulated and real quantum computers. First, for the subsequent circuit definition, evaluation and illustration, we first have to define auxiliary functions and import the corresponding modules from Qiskit and Numpy." - ] + ], + "metadata": {} }, { "cell_type": "code", "execution_count": 18, - "metadata": {}, - "outputs": [], "source": [ "import numpy as np # comments\n", "import random # \n", @@ -72,20 +70,20 @@ "from qiskit.ignis.mitigation.measurement import complete_meas_cal,CompleteMeasFitter \n", "from qiskit.visualization import plot_histogram \n", "from qiskit.test.mock import FakeVigo" - ] + ], + "outputs": [], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ " In general we employ four qubits to model the MPG. Two qubits are assigned to Alice and two qubits to Bob, while the state and move of the third qubit can be logically derieved from the first two. In a first step, we prepare code in order that these four qubits are brought into a state of superposition and entanglement. Further, we design the Cliffords U and V which conditionally change the architecture of the circuit and concomittant different rotations on the axis of the qubit with respect to a gamma, representing the randomily assigned row / column." - ] + ], + "metadata": {} }, { "cell_type": "code", "execution_count": 5, - "metadata": {}, - "outputs": [], "source": [ "# function that sets up the initial entanglement between Alice and Bob\n", "def share_bell_state(qc): \n", @@ -128,43 +126,20 @@ " qc.cz(a,b)\n", " qc.h(a)\n", " qc.h(b)" - ] + ], + "outputs": [], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "At this point, we can initialise the four qubits and bring them into the bell state (superposition of the first two qubits and entanglement between the first and third & second and fourth qubit), using the prior created share_bell_state function. Subsequently, we let the referee pick the random column and row with is passed to U and V, respectivly and visualize the as-is circuit. " - ] + ], + "metadata": {} }, { "cell_type": "code", "execution_count": 12, - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Alice and Bob, are assigned a column or row value of 1, 2, or 3\n", - "Alice assignment (column):\n", - "Bob assignment (row):\n", - "2\n", - " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", - "q9_0: ┤ H ├──■────────░───X──┤ H ├──────░─┤M├─────────\n", - " ├───┤ │ ░ │ ├───┤ ░ └╥┘┌─┐ \n", - "q9_1: ┤ H ├──┼────■───░───X──┤ I ├──────░──╫─┤M├──────\n", - " └───┘┌─┴─┐ │ ░ ┌───┐└───┘┌───┐ ░ ║ └╥┘┌─┐ \n", - "q9_2: ─────┤ X ├──┼───░─┤ Z ├──■──┤ H ├─░──╫──╫─┤M├───\n", - " └───┘┌─┴─┐ ░ ├───┤ │ ├───┤ ░ ║ ║ └╥┘┌─┐\n", - "q9_3: ──────────┤ X ├─░─┤ Z ├──■──┤ H ├─░──╫──╫──╫─┤M├\n", - " └───┘ ░ └───┘ └───┘ ░ ║ ║ ║ └╥┘\n", - "c9: 4/═════════════════════════════════════╩══╩══╩══╩═\n", - " 0 1 2 3 \n", - "haram\n" - ] - } - ], "source": [ "#We will ask the user to select the row and column on Alice and Bob's behalf.\n", "print(\"Alice and Bob, are assigned a column or row value of 1, 2, or 3\") \n", @@ -193,23 +168,45 @@ " \n", " magicsquare_circuit.draw(output='mpl')\n", " print(magicsquare_circuit) \n", - " print(\"haram\")\n", "else:\n", " print(\"please enter values of 1, 2, or 3\")" - ] + ], + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "Alice and Bob, are assigned a column or row value of 1, 2, or 3\n", + "Alice assignment (column):\n", + "Bob assignment (row):\n", + "2\n", + " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", + "q9_0: ┤ H ├──■────────░───X──┤ H ├──────░─┤M├─────────\n", + " ├───┤ │ ░ │ ├───┤ ░ └╥┘┌─┐ \n", + "q9_1: ┤ H ├──┼────■───░───X──┤ I ├──────░──╫─┤M├──────\n", + " └───┘┌─┴─┐ │ ░ ┌───┐└───┘┌───┐ ░ ║ └╥┘┌─┐ \n", + "q9_2: ─────┤ X ├──┼───░─┤ Z ├──■──┤ H ├─░──╫──╫─┤M├───\n", + " └───┘┌─┴─┐ ░ ├───┤ │ ├───┤ ░ ║ ║ └╥┘┌─┐\n", + "q9_3: ──────────┤ X ├─░─┤ Z ├──■──┤ H ├─░──╫──╫──╫─┤M├\n", + " └───┘ ░ └───┘ └───┘ ░ ║ ║ ║ └╥┘\n", + "c9: 4/═════════════════════════════════════╩══╩══╩══╩═\n", + " 0 1 2 3 \n", + "haram\n" + ] + } + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "However, at this point we still yet have to execute our circuit, however, prior to this step we must first define functions that extent the four qubits of Alice and Bob to the six game moves played and implement measures to determine the win / lose outcome of the played game. " - ] + ], + "metadata": {} }, { "cell_type": "code", "execution_count": 14, - "metadata": {}, - "outputs": [], "source": [ "#checks if the circuit output fulfills the winning criteria\n", "def check_result(results, col, row):\n", @@ -263,20 +260,20 @@ " print(\"------------------------------------------\") \n", " print(\"\\n\")\n", " return expected_states" - ] + ], + "outputs": [], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "Moreoever, for the purposes of explainability we describe and present the winning combinations for the selected row / column and from the four-qubit tangiable output the third qubits are computed. " - ] + ], + "metadata": {} }, { "cell_type": "code", "execution_count": 15, - "metadata": {}, - "outputs": [], "source": [ "# function explaining the computation of x3, y3 and presenting the corresponding winning moves\n", "def explain(result, col, row):\n", @@ -314,23 +311,31 @@ " print(\"------------------------------------------\") \n", " print(\"------------------------------------------\") \n", " print(\"\\n\")" - ] + ], + "outputs": [], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "Consequently, we can execute the circuit created above and directly observe the created results and moves and compare them with moves that solve the Mermin Peres Magic Square Game, we will oberseve that all of the possible moves that, both, Alice and Bob made fulfill the winning conditions of the MPE." - ] + ], + "metadata": {} }, { "cell_type": "code", "execution_count": 16, - "metadata": {}, + "source": [ + "job = execute(magicsquare_circuit, BasicAer.get_backend('qasm_simulator'), optimization_level=3, shots=8192) # run the job simulation\n", + "results = job.result().get_counts()\n", + "print(results)\n", + "expected_states = check_result(results, alpha, beta)\n", + "print(expected_states)" + ], "outputs": [ { - "name": "stdout", "output_type": "stream", + "name": "stdout", "text": [ "{'0010': 1067, '1100': 1000, '0110': 977, '1000': 1026, '0101': 1001, '0001': 1039, '1011': 1035, '1111': 1047}\n", "Resultant quantum circuit solution:\n", @@ -554,24 +559,17 @@ ] } ], - "source": [ - "job = execute(magicsquare_circuit, BasicAer.get_backend('qasm_simulator'), optimization_level=3, shots=8192) # run the job simulation\n", - "results = job.result().get_counts()\n", - "print(results)\n", - "expected_states = check_result(results, alpha, beta)\n", - "print(expected_states)" - ] + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "Indeed, using an ideal quantum simulator, a winning probability of 100% is achieved, which exceeds the highest classical winning probability of 88.99% Beyond that, however, it is desirable to quantify the results on a real IBM quantum computer, which will be realized in the last section." - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "# Mathmatical background of the computation\n", "\n", @@ -595,37 +593,37 @@ "$$ Quantum Solution MPG = Initial Vector * Dynamic Matrix(row, column)$$\n", "\n", "Starting with the first part of the equation, the intial vector, of the MPG circuit is defined by quantum gates as follows:\n" - ] + ], + "metadata": {} }, { "cell_type": "code", "execution_count": 11, - "metadata": {}, + "source": [ + "q = QuantumRegister(4) \n", + "c = ClassicalRegister(4) \n", + "qc = QuantumCircuit(q, c) \n", + "share_bell_state(qc)\n", + "qc.draw(output='mpl')\n" + ], "outputs": [ { + "output_type": "execute_result", "data": { - "image/png": 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+ "image/svg+xml": "\n\n\n\n \n \n \n \n 2021-05-19T14:15:25.228767\n image/svg+xml\n \n \n Matplotlib v3.3.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n" }, - "execution_count": 11, "metadata": {}, - "output_type": "execute_result" + "execution_count": 11 } ], - "source": [ - "q = QuantumRegister(4) \n", - "c = ClassicalRegister(4) \n", - "qc = QuantumCircuit(q, c) \n", - "share_bell_state(qc)\n", - "qc.draw(output='mpl')\n" - ] + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "This initial vector is generated, as illustrated above, by the application of two Hadamard gates to the first and second qubits and CNOT gates, with the first and second qubits serving as the control qubit and the third and fourth qubits, respectively, serving as the target qubit. Mathematically, these operations can be described by the following equation: \n", "\n", @@ -634,19 +632,19 @@ "
\n", "\n", "However, prior to inserting the corresponding matrices into this equation, we introduce the relevant matrices needed for the computation. " - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "The very starting vector state of the MPG circuit, prior to any quantum gate interaction, corresponds to the Dirac bracket |0000> and is hence described by the following vector: \n", "$$ InitialQubits= {\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}$$" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "In terms of the gates, these are the mathmatical represetations of the relevant gates illustrated above. \n", "\n", @@ -672,20 +670,20 @@ "
\n", "The \"i\" in CNOT(i, i+2) refers to the control qubit in the first instance and to the target qubit in the second instance under the condition that there are at least two qubits in between such as $CNOT_(1,3)$\n", "\n" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "Now we can recall the equation of the initial vectors and insert the matrices:\n", "\n", "$$ Initial Vector = (I ⨂ CNOT_{0|2}) ⨂ (CNOT_{1|3} ⨂ I) * (I ⨂ I ⨂ H ⨂ H) * InitialQubits $$\n" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "$$\n", "Inital Vector = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} *\n", @@ -705,11 +703,11 @@ "*\n", "{\\begin{pmatrix} 1 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\ 0\\end{pmatrix}}\n", "$$" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "$$ \n", "InitialVector = \\begin{bmatrix}\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{bmatrix}\\\n", @@ -727,19 +725,19 @@ "$$\n", "= \\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\ 0\\\\ \\frac 1 {2}\\end{pmatrix}\n", "$$" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "This resulting vector holds the relevant qubit states that can be further considered and modified in the dynmaic part of the circuit so that valid winning condition can be created, these Qubit states are:\n", "$$= \\frac 1 {2} (|0000\\rangle+|0101\\rangle+|1010\\rangle+|1111\\rangle)$$\n" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "Furthermore, when analyzing the initial vector, it becomes apparent that when multiplying any matrix by it, only four relevenate cells per row in the multiplier remain open for consideration, since the others are all multiplied by 0 and thereby likewise drop of with 0% probability as a result. Thus, we can eliminate the majority of the entries whatsoever. As an example we have prepared the matrix below, and denoted all entries with an i, that have to be computed and all others that result in any case with 0 with -.\n", "\n", @@ -774,46 +772,46 @@ "= \\frac 1 {\\sqrt{8}}\n", "$$\n", "And indeed, this idea is very powerful because it lets us visually see what outcomes may emerge as well asa better idea of which outcomes may fall off and which may be a win in the MPG. As long as postive result emerges the MPG is won, no matter the probability of that play occuring" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "Now, in order to consider the later, dynamic, part of the equation we will have to select an example. In this case we have selected row: 1 and column: 1. The corresponding quantum circuit is as follows:" - ] + ], + "metadata": {} }, { "cell_type": "code", "execution_count": 14, - "metadata": {}, + "source": [ + "q = QuantumRegister(4) \n", + "c = ClassicalRegister(4) \n", + "qc = QuantumCircuit(q, c) \n", + "qc.barrier()\n", + "U(qc,1) \n", + "V(qc,1)\n", + "qc.draw(output='mpl')" + ], "outputs": [ { + "output_type": "execute_result", "data": { - "image/png": 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", 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+ "image/svg+xml": "\n\n\n\n \n \n \n \n 2021-05-19T14:16:22.330352\n image/svg+xml\n \n \n Matplotlib v3.3.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n" }, - "execution_count": 14, "metadata": {}, - "output_type": "execute_result" + "execution_count": 14 } ], - "source": [ - "q = QuantumRegister(4) \n", - "c = ClassicalRegister(4) \n", - "qc = QuantumCircuit(q, c) \n", - "qc.barrier()\n", - "U(qc,1) \n", - "V(qc,1)\n", - "qc.draw(output='mpl')" - ] + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "The translation of these circuit gate operations is in turn equivalent to the following mathematical matrice multiplications:\n", "$$ DynamicMatrix(1, 1) = H ⨂ H ⨂ I ⨂ H $$\n", @@ -853,18 +851,18 @@ " \\\\\\ \n", "0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & -\\frac 1 {\\sqrt{8}} & \\frac 1 {\\sqrt{8}} & 0 & 0 & \\frac 1 {\\sqrt{8}} & -\\frac 1 {\\sqrt{8}} \\\\\\end{bmatrix}\\ \n", "$$" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "Now, as we have both, the inital vector and the dynamic matrix, we can compute the winning plays for Alice and Bob with selected column and row of 1. In prinicple we can apply the rule of the i-rows, that we have established above, to this matrix-vector multiplication, however, in this case for the sake of completness we let all entries remain. But feel free to check the rule!" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "$$\n", "Winning Plays = DynamicMatrix * InitialVector\n", @@ -906,24 +904,23 @@ "$$\n", "= \\begin{pmatrix} \\frac 1 {\\sqrt{8}} \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\\\frac 1 {\\sqrt{8}} \\\\0 \\\\ -\\frac 1 {\\sqrt{8}} \\\\0 \\\\0 \\\\\\frac 1 {\\sqrt{8}} \\\\ 0\\\\ -\\frac 1 {\\sqrt{8}} \\end{pmatrix}\n", "$$\n" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "Finally, this vector contains the qubits that can be played to win the game with probability 1, these are:\n", "\n", "$$= \\frac 1 {\\sqrt{8}} (|0000\\rangle+|0100\\rangle+|0101\\rangle+|01111\\rangle+|1000\\rangle)-|1011\\rangle)+|1101\\rangle)-|1111\\rangle)$$\n", "\n", "To validate these mathematical results, we will run the quantum circuit in one go and compare the respective qubit results, we can formalize and measure the rate of the winning condition for multiple executions with the state accuracy, which compares the expected, the ideal, states with the resultant states of the circuit and measures deviations: " - ] + ], + "metadata": {} }, { "cell_type": "code", "execution_count": null, - "metadata": {}, - "outputs": [], "source": [ "# function to calculate state accuracy\n", "def get_state_accuracy(counts, expected_states, shots): \n", @@ -933,33 +930,20 @@ " expected_counts = expected_counts + counts[state]\n", " state_accuracy = expected_counts / shots\n", " return str(state_accuracy*100)+\"%\"" - ] + ], + "outputs": [], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "For row:=1 and column:=1 the following circuit is generated: " - ] + ], + "metadata": {} }, { "cell_type": "code", "execution_count": 16, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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", - "image/svg+xml": "\n\n\n\n \n \n \n \n 2021-05-19T14:16:45.991858\n image/svg+xml\n \n \n Matplotlib v3.3.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n", - "text/plain": [ - "
" - ] - }, - "execution_count": 16, - "metadata": {}, - "output_type": "execute_result" - } - ], "source": [ "q = QuantumRegister(4) \n", "c = ClassicalRegister(4) \n", @@ -970,16 +954,37 @@ "V(qc,1)\n", "qc.measure(q,c)\n", "qc.draw(output='mpl')" - ] + ], + "outputs": [ + { + "output_type": "execute_result", + "data": { + "text/plain": [ + "
" + ], + "image/png": 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", + "image/svg+xml": "\n\n\n\n \n \n \n \n 2021-05-19T14:16:45.991858\n image/svg+xml\n \n \n Matplotlib v3.3.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n" + }, + "metadata": {}, + "execution_count": 16 + } + ], + "metadata": {} }, { "cell_type": "code", "execution_count": 17, - "metadata": {}, + "source": [ + "backend = BasicAer.get_backend('qasm_simulator')\n", + "results = execute(qc, backend, optimization_level=3, shots=8192).result().get_counts()\n", + "expected_states = check_result(results, 1, 1)\n", + "state_accuracy = get_state_accuracy(results,expected_states,8192)\n", + "print(\"Accuracy: \"+state_accuracy)" + ], "outputs": [ { - "name": "stdout", "output_type": "stream", + "name": "stdout", "text": [ "Resultant quantum circuit solution:\n", "\n", @@ -1202,24 +1207,17 @@ ] } ], - "source": [ - "backend = BasicAer.get_backend('qasm_simulator')\n", - "results = execute(qc, backend, optimization_level=3, shots=8192).result().get_counts()\n", - "expected_states = check_result(results, 1, 1)\n", - "state_accuracy = get_state_accuracy(results,expected_states,8192)\n", - "print(\"Accuracy: \"+state_accuracy)" - ] + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "Evidently, the results of the mathematical calculation match those of the circuit and a win rate of 100% is achieved!" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "------------------------------------------------------------------------------------------------------------------------------------------------------------------\n", "To reproduce these operations for other configurations, it is recommended to use the Unitary simulator - follow this simple guide:\n", @@ -1236,32 +1234,39 @@ "result = execute(qc, BasicAer.get_backend('unitary_simulator')).result().get_unitary(qc)
\n", "array_to_latex(result, pretext=\"\\\\text{Circuit = }\\n\") \n", "------------------------------------------------------------------------------------------------------------------------------------------------------------------" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "# Circuit Performance on Real Quantum Devices" - ] + ], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "Thus far all executions of the Quantum circuits were performed on ideal quantum simulators, rather than actual quantum hardware. Real quantum computers have noise, decoherence and other periphery hardware issues, which can reduce the performance and fidelity of the executions. Hence, in order to ascertain the actucal performance and not only the theoretical we will execute our circuit on real quantum computers as well as simulators of real computers that include the above mentioned errors alike. \n", "\n", "We start off with the non-ideal simulators of the real quantum computers, so called FakeDevices. The use of FakeVigo, simualating the quantum Computer Vigo yields the following results." - ] + ], + "metadata": {} }, { "cell_type": "code", "execution_count": 24, - "metadata": {}, + "source": [ + "job = execute(magicsquare_circuit, FakeVigo(), optimization_level=3, shots=8192) # run the job simulation\n", + "results = job.result().get_counts()\n", + "expected_states = check_result(results, alpha, beta)\n", + "state_accuracy = get_state_accuracy(results,expected_states,8192)\n", + "print(state_accuracy)" + ], "outputs": [ { - "name": "stdout", "output_type": "stream", + "name": "stdout", "text": [ "Resultant quantum circuit solution:\n", "\n", @@ -1612,66 +1617,65 @@ ] } ], - "source": [ - "job = execute(magicsquare_circuit, FakeVigo(), optimization_level=3, shots=8192) # run the job simulation\n", - "results = job.result().get_counts()\n", - "expected_states = check_result(results, alpha, beta)\n", - "state_accuracy = get_state_accuracy(results,expected_states,8192)\n", - "print(state_accuracy)" - ] + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "We can oberserve that simulated noisy devices outperform classical systems and thus indicate that real quantum computers are capable of doing the same. Now in order to evaluate the performance of real devices we select the ibmq_athens computer to benchmark the MPG. Please note that you will have to add your own API token from the IBM Quantum website (https://quantum-computing.ibm.com) so that this cell executes" - ] + ], + "metadata": {} }, { "cell_type": "code", "execution_count": 2, - "metadata": {}, + "source": [ + "IBMQ.save_account('API_Key') # retrieve from your IBM-Account: https://quantum-computing.ibm.com" + ], "outputs": [ { - "name": "stderr", "output_type": "stream", + "name": "stderr", "text": [ "configrc.store_credentials:WARNING:2021-05-19 14:15:02,195: Credentials already present. Set overwrite=True to overwrite.\n" ] } ], - "source": [ - "IBMQ.save_account('API_Key') # retrieve from your IBM-Account: https://quantum-computing.ibm.com" - ] + "metadata": {} }, { "cell_type": "code", "execution_count": 3, - "metadata": {}, + "source": [ + "# load IBM Quantum Experience account \n", + "IBMQ.load_account()\n", + "provider = IBMQ.get_provider('ibm-q') # get provider" + ], "outputs": [ { - "name": "stderr", "output_type": "stream", + "name": "stderr", "text": [ "/Users/david.drexlin@ibm.com/opt/anaconda3/lib/python3.8/site-packages/qiskit/providers/ibmq/ibmqfactory.py:192: UserWarning: Timestamps in IBMQ backend properties, jobs, and job results are all now in local time instead of UTC.\n", " warnings.warn('Timestamps in IBMQ backend properties, jobs, and job results '\n" ] } ], - "source": [ - "# load IBM Quantum Experience account \n", - "IBMQ.load_account()\n", - "provider = IBMQ.get_provider('ibm-q') # get provider" - ] + "metadata": {} }, { "cell_type": "code", "execution_count": 69, - "metadata": {}, + "source": [ + "job = execute(magicsquare_circuit, least_busy(provider.backends(simulator=False)),optimization_level=3, shots=8192)\n", + "results = job.result().get_counts()\n", + "expected_states = check_result(results, alpha, beta)\n", + "state_accuracy = get_state_accuracy(results,expected_states,8192)" + ], "outputs": [ { - "name": "stdout", "output_type": "stream", + "name": "stdout", "text": [ "Resultant quantum circuit solution:\n", "\n", @@ -2021,86 +2025,113 @@ ] } ], - "source": [ - "job = execute(magicsquare_circuit, least_busy(provider.backends(simulator=False)),optimization_level=3, shots=8192)\n", - "results = job.result().get_counts()\n", - "expected_states = check_result(results, alpha, beta)\n", - "state_accuracy = get_state_accuracy(results,expected_states,8192)" - ] + "metadata": {} }, { "cell_type": "code", "execution_count": 70, - "metadata": {}, + "source": [ + "print(state_accuracy)" + ], "outputs": [ { - "name": "stdout", "output_type": "stream", + "name": "stdout", "text": [ "68.51806640625%\n" ] } ], - "source": [ - "print(state_accuracy)" - ] + "metadata": {} }, { "cell_type": "code", "execution_count": 28, - "metadata": {}, - "outputs": [], "source": [ "backend = provider.get_backend(\"ibmq_athens\")" - ] + ], + "outputs": [], + "metadata": {} }, { "cell_type": "code", "execution_count": 395, - "metadata": {}, - "outputs": [], "source": [ "job = execute(magicsquare_circuit, backend,optimization_level=3, shots=8192) # run the job simulation with the highest compilter optimsation\n", "results = job.result().get_counts()\n", "expected_states = check_result(results, alpha, beta)\n", "state_accuracy = get_state_accuracy(results,expected_states,8192)" - ] + ], + "outputs": [], + "metadata": {} }, { "cell_type": "code", "execution_count": 396, - "metadata": {}, + "source": [ + "state_accuracy" + ], "outputs": [ { + "output_type": "execute_result", "data": { "text/plain": [ "'96.69189453125%'" ] }, - "execution_count": 396, "metadata": {}, - "output_type": "execute_result" + "execution_count": 396 } ], - "source": [ - "state_accuracy" - ] + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "As outlined above, the MPG circuit outperforms the classical circuit by about 7.8% (88.89% vs. 96.69%) on a real, noisy, quantum computer. Further optimising can be achieved via measurement error mitigation as can be further seen in this [notebook](https://qiskit.org/textbook/ch-quantum-hardware/measurement-error-mitigation.html)." - ] + ], + "metadata": {} }, { "cell_type": "code", "execution_count": 32, - "metadata": {}, + "source": [ + "# measurement error mitigation on backend 1\n", + "results_backend1_mem = []\n", + "\n", + "# normal execution\n", + "noisy_counts = execute(magicsquare_circuit, backend=backend, shots=8192, optimization_level=0).result().get_counts() # get results\n", + "expected_states = check_result(noisy_counts, alpha, beta)\n", + "state_accuracy = get_state_accuracy(noisy_counts,expected_states,8192)\n", + "results_backend1_mem.append((\"noisy\", noisy_counts, state_accuracy))\n", + "\n", + "# measurement error mitigation\n", + "qr = QuantumRegister(4)\n", + "meas_calibs, state_labels = complete_meas_cal(qr=qr, circlabel='mcal') # setup calibration measures on basis states\n", + "\n", + "cal_results = execute(meas_calibs, backend=backend, shots=8192).result() # execute calibration measures\n", + "meas_fitter = CompleteMeasFitter(cal_results, state_labels, circlabel='mcal')\n", + "meas_filter = meas_fitter.filter # get the filter object from calibration\n", + "\n", + "mitigated_counts = meas_filter.apply(noisy_counts) # apply filter to get mitigated results\n", + "expected_states = check_result(mitigated_counts, alpha, beta)\n", + "state_accuracy = get_state_accuracy(mitigated_counts,expected_states,8192)\n", + "results_backend1_mem.append((\"mitigated\", mitigated_counts, state_accuracy))\n", + "\n", + "# compare state accuracies\n", + "print(\"State Accuracy \\n\"+\n", + " \"\\n\".join([str(label)+': '+str(state_accuracy) for label, counts, state_accuracy in results_backend1_mem]))\n", + "\n", + "# plot results\n", + "plot_histogram([counts for optim_level, counts, state_accuracy in results_backend1_mem],\n", + " legend=[label for label, counts, state_accuracy in results_backend1_mem],\n", + " title=backend.name(),\n", + " bar_labels=False)" + ], "outputs": [ { - "name": "stdout", "output_type": "stream", + "name": "stdout", "text": [ "Resultant quantum circuit solution:\n", "\n", @@ -2798,77 +2829,45 @@ ] }, { + "output_type": "execute_result", "data": { - "image/png": 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", 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"metadata": {}, - "output_type": "execute_result" + "execution_count": 32 } ], - "source": [ - "# measurement error mitigation on backend 1\n", - "results_backend1_mem = []\n", - "\n", - "# normal execution\n", - "noisy_counts = execute(magicsquare_circuit, backend=backend, shots=8192, optimization_level=0).result().get_counts() # get results\n", - "expected_states = check_result(noisy_counts, alpha, beta)\n", - "state_accuracy = get_state_accuracy(noisy_counts,expected_states,8192)\n", - "results_backend1_mem.append((\"noisy\", noisy_counts, state_accuracy))\n", - "\n", - "# measurement error mitigation\n", - "qr = QuantumRegister(4)\n", - "meas_calibs, state_labels = complete_meas_cal(qr=qr, circlabel='mcal') # setup calibration measures on basis states\n", - "\n", - "cal_results = execute(meas_calibs, backend=backend, shots=8192).result() # execute calibration measures\n", - "meas_fitter = CompleteMeasFitter(cal_results, state_labels, circlabel='mcal')\n", - "meas_filter = meas_fitter.filter # get the filter object from calibration\n", - "\n", - "mitigated_counts = meas_filter.apply(noisy_counts) # apply filter to get mitigated results\n", - "expected_states = check_result(mitigated_counts, alpha, beta)\n", - "state_accuracy = get_state_accuracy(mitigated_counts,expected_states,8192)\n", - "results_backend1_mem.append((\"mitigated\", mitigated_counts, state_accuracy))\n", - "\n", - "# compare state accuracies\n", - "print(\"State Accuracy \\n\"+\n", - " \"\\n\".join([str(label)+': '+str(state_accuracy) for label, counts, state_accuracy in results_backend1_mem]))\n", - "\n", - "# plot results\n", - "plot_histogram([counts for optim_level, counts, state_accuracy in results_backend1_mem],\n", - " legend=[label for label, counts, state_accuracy in results_backend1_mem],\n", - " title=backend.name(),\n", - " bar_labels=False)" - ] + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "The noisy could be reduced in this experiment by further 4%, however, the orginal results could not be reconstricuted and thus attained or surpassed. At this point, if running this notebook on a shared system (e.g. mybinder.org). Remember to log off from your IBM Q account in case you used it for execution on real devices by running the following cell:" - ] + ], + "metadata": {} }, { "cell_type": "code", "execution_count": null, - "metadata": {}, - "outputs": [], "source": [ "IBMQ.disable_account()\n", "IBMQ.delete_account()" - ] + ], + "outputs": [], + "metadata": {} }, { "cell_type": "markdown", - "metadata": {}, "source": [ "#### Finally, this conclues our experiments on the Mermin Peres Magic Sqaure Game. We elaborated the quantum architecure of the game, presented its scheme of operation and quantified the results on real quantum computers. \n", "\n", "_Dr. Jan-Rainer Lahmann - David Drexlin_" - ] + ], + "metadata": {} } ], "metadata": { From 40a3430787ecca3b81c4001c5499df5c253620d1 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Tue, 24 Aug 2021 22:09:06 +0200 Subject: [PATCH 28/30] fix: typos --- "Mermin\342\200\223Peres-Game.ipynb" | 125 +++++++++++++-------------- 1 file changed, 61 insertions(+), 64 deletions(-) diff --git "a/Mermin\342\200\223Peres-Game.ipynb" "b/Mermin\342\200\223Peres-Game.ipynb" index 1a5509b..585f545 100644 --- "a/Mermin\342\200\223Peres-Game.ipynb" +++ "b/Mermin\342\200\223Peres-Game.ipynb" @@ -10,11 +10,11 @@ { "cell_type": "markdown", "source": [ - "Telepathy, the ability to read the mind of other humans has gripped men since the begining of times. In modern times, however, there are no technolgies and methodologies to read the minds of humans whatsoever and the field is largely conisdered to be pseudoscientific. However, in nature there are at the quantum mechanical level, effects which in an abstracted way mimic and model forms of telepathy. In practice, the application of the quantum mechanical principlies of qunatum entanglment and quantum superposition enable the instantaneous exchange of qunatum information on the collapse of the wave function, without an expplicilt information exchange, this effecr persits even when seperated by many light years where classical communcation would take more than a life time. \n", + "Telepathy, the ability to read the mind of other humans, has gripped men since the beginning of times. In modern times, however, there are no technologies and methodologies to read the minds of humans whatsoever and the field is largely considered to be pseudoscientific. However, in nature there are, at the quantum mechanical level, effects which in an abstracted way mimic and model forms of telepathy. In practice, the application of the quantum mechanical principles of quantum entanglement and quantum superposition enable the instantaneous exchange of quantum information on the collapse of the wave function, without an explicit information exchange, this effect persist even when separated by many light years when classical communication would take more than a lifetime. \n", "\n", - "The application of these principles to problems has developed solutions. which which are considered to be unsolvable or not trivial to solve in non-quantum systems. Within game theory, and for games the with incomplete information this has given players the telepathic ability to derieve information without communcation from the game due to superposition and entanglement, this fact has been dubbed 'quantum pseudo-telepathy'.\n", + "The application of these principles to problems has developed solutions. which are considered to be unsolvable or not trivial to solve in non-quantum systems. Within game theory, and for games the with incomplete information this has given players the telepathic ability to derive information without communication from the game due to superposition and entanglement, this fact has been dubbed 'quantum pseudo-telepathy'.\n", "\n", - "For further illustration of this \"quantum psuedo-telepathic\" ability let us consider the following problem known under the term the Mermin Peres Magic Square Game (MPG) as visualized below: Two players, nicknamed Alice and Bob, play a cooperative game and are bindingly not allowed to communicate during the game with each other. Both player are assigned a random row and column in a 3x3 grid where they either place zero or two tiles in their respective row / colum. Indivually, they can place their tiles in any or naught of the three possible cells. However, and most curucaiylly there must be unity with respect to the placment of the tiles in the intersecting cell, of the row and the column. Only either one tile of either player may and must be placed there. In consequence and as we will explore further, due to the fact that there is no communication possible, there exist no determinsitc strategy with which the game can be won with a probability of 1, even with a prior collusive agreement to play certain tiles in specific cells, for certain row / column combinations. \n" + "For further illustration of this \"quantum psuedo-telepathic\" ability let us consider the following problem known under the term the Mermin-Peres Magic Square Game (MPG) as visualized below: Two players, nicknamed Alice and Bob, play a cooperative game and are bindingly not allowed to communicate during the game with each other. Both players are assigned a random row and column in a 3x3 grid, where they either place zero or two tiles in their respective row / column. Individually, they can place their tiles in any or naught of the three possible cells. However, and most crucially, there must be unity with respect to the placement of the tiles in the intersecting cell, of the row and the column. Only either one tile of either player may and must be placed there. In consequence and as we will explore further, due to the fact that there is no communication possible, there exist no deterministic strategy with which the game can be won with a probability of 1, even with a prior collusive agreement to play certain tiles in specific cells, for certain row / column combinations. " ], "metadata": {} }, @@ -28,18 +28,14 @@ { "cell_type": "markdown", "source": [ - "More formally, rather than placing 'graphical' tiles (as seen above), a conceptually equivalent and necessary modifaction for the mathmtical proof of the unsolvable nature of the MPG is the placement of digits of +1 and -1 by Bob and Alice, respectivly instead of 'A' and 'B' as well as. Hereafter, the tile B will be replaced by +1 and tile A by -1, another [recent publication further details on the scheme of operation] (https://medium.com/qiskit/this-proof-demonstrates-a-quantum-advantage-even-for-noisy-quantum-computers-b44a738801ad).\n", + "More formally, rather than placing 'graphical' tiles (as seen above), a conceptually equivalent and necessary modification for the mathematical proof of the unsolvable nature of the MPG is the placement of digits of +1 and -1 by Bob and Alice, respectively instead of 'A' and 'B' as well as. Hereafter, the tile B will be replaced by +1 and tile A by -1, another [recent publication further details on the scheme of operation] (https://medium.com/qiskit/this-proof-demonstrates-a-quantum-advantage-even-for-noisy-quantum-computers-b44a738801ad).\n", "\n", - "Mathmatically, the rules that either zero or two tiles must be placed can be mathmatically translated into the fact \n", - "\n", - " game rules can be stated \n", - "\n", - "we now formualte three conditions which must be met, so that the game is won:\n", + "When we substitute the tiles 'A' and 'B' with +1 and -1 a mathematical description of the winning conditions becomes feasible, for Bob the product of the row must be equal to +1 and the product of the column must be -1 for Alice: \n", "\n", "1. $\\overset{3}{\\underset{i = 1}{\\Pi}} m_{i,j} = -1 \\quad \\forall j \\in{1,2,3} $\n", "2. $\\overset{3}{\\underset{j = 1}{\\Pi}} m_{i,j} = +1 \\quad \\forall i \\in{1,2,3} $\n", "\n", - "3. For the third rule, and the fact of the intersection between the column and row a confict is bound to emerge as the $m_{i,j}$ variables are commuting and thus can't describe both winning conditions:\n", + "3. For the third rule, and the fact of the intersection between the column and row a conflict is bound to emerge as the $m_{i,j}$ variables are commuting and thus can't describe both winning conditions:\n", "- $\\overset{3}{\\underset{j = 1}{\\Pi}}\\overset{3}{\\underset{i = 1}{\\Pi}} m_{i,j} = -1$\n", "- $\\overset{3}{\\underset{i = 1}{\\Pi}}\\overset{3}{\\underset{j = 1}{\\Pi}} m_{i,j} = +1$ \n", "- $-1 = \\overset{3}{\\underset{i,j = 1}{\\Pi}}m_{i,j} = +1 $\n", @@ -51,13 +47,13 @@ { "cell_type": "markdown", "source": [ - "To the end of implemenation, the Python Qiskit framework was employed. Subsequently, this notebook will introduce the quantum circuit solving the MPG game, provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game and quantify it's performance on both simulated and real quantum computers. First, for the subsequent circuit definition, evaluation and illustration, we first have to define auxiliary functions and import the corresponding modules from Qiskit and Numpy." + "To the end of implementation, the Python Qiskit framework was employed. Subsequently, this notebook will introduce the quantum circuit solving the MPG game, provide a step-by-step walkthrough of the mathematical operation applied during an exemplary game and quantify its performance on both simulated and real quantum computers. First, for the subsequent circuit definition, evaluation and illustration, we first have to define auxiliary functions and import the corresponding modules from Qiskit and Numpy." ], "metadata": {} }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 1, "source": [ "import numpy as np # comments\n", "import random # \n", @@ -77,13 +73,13 @@ { "cell_type": "markdown", "source": [ - " In general we employ four qubits to model the MPG. Two qubits are assigned to Alice and two qubits to Bob, while the state and move of the third qubit can be logically derieved from the first two. In a first step, we prepare code in order that these four qubits are brought into a state of superposition and entanglement. Further, we design the Cliffords U and V which conditionally change the architecture of the circuit and concomittant different rotations on the axis of the qubit with respect to a gamma, representing the randomily assigned row / column." + " In general, we employ four qubits to model the MPG. Two qubits are assigned to Alice and two qubits to Bob, while the state and move of the third qubit can be logically derived from the first two. In a first step, we prepare code in order that these four qubits are brought into a state of superposition and entanglement. Further, we design the Cliffords U and V which conditionally change the architecture of the circuit and concomitant different rotations on the axis of the qubit with respect to a gamma, representing the randomly assigned row / column." ], "metadata": {} }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 2, "source": [ "# function that sets up the initial entanglement between Alice and Bob\n", "def share_bell_state(qc): \n", @@ -133,13 +129,13 @@ { "cell_type": "markdown", "source": [ - "At this point, we can initialise the four qubits and bring them into the bell state (superposition of the first two qubits and entanglement between the first and third & second and fourth qubit), using the prior created share_bell_state function. Subsequently, we let the referee pick the random column and row with is passed to U and V, respectivly and visualize the as-is circuit. " + "At this point, we can initialize the four qubits and bring them into the bell state (superposition of the first two qubits and entanglement between the first and third & second and fourth qubit), using the prior created share_bell_state function. Subsequently, we let the referee pick the random column and row with is passed to U and V, respectively, and visualize the as-is circuit. " ], "metadata": {} }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 3, "source": [ "#We will ask the user to select the row and column on Alice and Bob's behalf.\n", "print(\"Alice and Bob, are assigned a column or row value of 1, 2, or 3\") \n", @@ -179,19 +175,18 @@ "Alice and Bob, are assigned a column or row value of 1, 2, or 3\n", "Alice assignment (column):\n", "Bob assignment (row):\n", - "2\n", - " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", - "q9_0: ┤ H ├──■────────░───X──┤ H ├──────░─┤M├─────────\n", - " ├───┤ │ ░ │ ├───┤ ░ └╥┘┌─┐ \n", - "q9_1: ┤ H ├──┼────■───░───X──┤ I ├──────░──╫─┤M├──────\n", - " └───┘┌─┴─┐ │ ░ ┌───┐└───┘┌───┐ ░ ║ └╥┘┌─┐ \n", - "q9_2: ─────┤ X ├──┼───░─┤ Z ├──■──┤ H ├─░──╫──╫─┤M├───\n", - " └───┘┌─┴─┐ ░ ├───┤ │ ├───┤ ░ ║ ║ └╥┘┌─┐\n", - "q9_3: ──────────┤ X ├─░─┤ Z ├──■──┤ H ├─░──╫──╫──╫─┤M├\n", - " └───┘ ░ └───┘ └───┘ ░ ║ ║ ║ └╥┘\n", - "c9: 4/═════════════════════════════════════╩══╩══╩══╩═\n", - " 0 1 2 3 \n", - "haram\n" + "3\n", + " ┌───┐ ░ ┌───┐ ░ ┌─┐ \n", + "q0_0: ┤ H ├──■────────░───■──┤ H ├─░─┤M├─────────\n", + " ├───┤ │ ░ ┌─┴─┐├───┤ ░ └╥┘┌─┐ \n", + "q0_1: ┤ H ├──┼────■───░─┤ X ├┤ I ├─░──╫─┤M├──────\n", + " └───┘┌─┴─┐ │ ░ └───┘└───┘ ░ ║ └╥┘┌─┐ \n", + "q0_2: ─────┤ X ├──┼───░───X────────░──╫──╫─┤M├───\n", + " └───┘┌─┴─┐ ░ │ ░ ║ ║ └╥┘┌─┐\n", + "q0_3: ──────────┤ X ├─░───X────────░──╫──╫──╫─┤M├\n", + " └───┘ ░ ░ ║ ║ ║ └╥┘\n", + "c0: 4/════════════════════════════════╩══╩══╩══╩═\n", + " 0 1 2 3 \n" ] } ], @@ -200,7 +195,7 @@ { "cell_type": "markdown", "source": [ - "However, at this point we still yet have to execute our circuit, however, prior to this step we must first define functions that extent the four qubits of Alice and Bob to the six game moves played and implement measures to determine the win / lose outcome of the played game. " + "However, at this point we still yet have to execute our circuit, however, prior to this step we must first define functions that extend the four qubits of Alice and Bob to the six game moves played and implement measures to determine the win / lose outcome of the played game. " ], "metadata": {} }, @@ -267,7 +262,7 @@ { "cell_type": "markdown", "source": [ - "Moreoever, for the purposes of explainability we describe and present the winning combinations for the selected row / column and from the four-qubit tangiable output the third qubits are computed. " + "Moreover, for the purposes of explainability we describe and present the winning combinations for the selected row / column and from the four-qubit tangible output the third qubits are computed." ], "metadata": {} }, @@ -318,7 +313,7 @@ { "cell_type": "markdown", "source": [ - "Consequently, we can execute the circuit created above and directly observe the created results and moves and compare them with moves that solve the Mermin Peres Magic Square Game, we will oberseve that all of the possible moves that, both, Alice and Bob made fulfill the winning conditions of the MPE." + "Consequently, we can execute the circuit created above and directly observe the created results and moves and compare them with moves that solve the Mermin-Peres Magic Square Game, we will observe that all the possible moves that, both, Alice and Bob made fulfill the winning conditions of the MPE." ], "metadata": {} }, @@ -573,15 +568,15 @@ "source": [ "# Mathmatical background of the computation\n", "\n", - "As we have demonstrated above, the quantum circuit solves the Mirmin-Peres game (MPG) on simulators with a higher probability than with any classical strategy on ordinary computers possible. To shed some light on how and why this works, we present and explain the mathematical operations that lead to the solution of this game. In particular, we will first present a high-level equation describing the mathematical rationale and then we will take a look at each element of this equation explain it step by step.\n", + "As we have demonstrated above, the quantum circuit solves the Mermin-Peres game (MPG) on simulators with a higher probability than with any classical strategy on ordinary computers possible. To shed some light on how and why this works, we present and explain the mathematical operations that lead to the solution of this game. In particular, we will first present a high-level equation describing the mathematical rationale, and then we will take a look at each element of this equation to explain it step by step.\n", "\n", "\n", - "Recall that the quantum circuit has to play different moves for each different row/column instances of the MPG being selected, since there is no classical strategy that can win the game with probability = 1. Thus, to win with probability 1, the quantum circuit dynamically adapts to the input of each column and row played. Nevertheless, there is indeed one part of the circuit that remains static and is responsible for generating a matching qubit state, that entangles the qubits of Alice and Bob. As this part comes prior to any adaptation we will start off by focusing on that part. \n", + "Recall that the quantum circuit has to play different moves for each different row/column instances of the MPG being selected, since there is no classical strategy that can win the game with probability = 1. Thus, to win with probability 1, the quantum circuit dynamically adapts to the input of each column and row played. Nevertheless, there is indeed one part of the circuit that remains static and is responsible for generating a matching qubit state, that entangles the qubits of Alice and Bob. As this part comes prior to any adaptation, we will start off by focusing on that part. \n", "\n", "![title](Mirmin-Peres-Game/Pauli_Operators.png)\n", "\n", "\n", - "These operators can then be translated into the [following quantum gates] (https://arxiv.org/pdf/1904.01502.pdf) for the r(spective row / column selected. To further understand why these Pauli operators are translated to the gates, see.: [Quantum advantage in noisy flat circuits](https://arxiv.org/pdf/1904.01502.pdf) \n", + "These operators can then be translated into the [following quantum gates] (https://arxiv.org/pdf/1904.01502.pdf) for the respective row / column selected. To further understand why these Pauli operators are translated to the gates, see.: [Quantum advantage in noisy flat circuits](https://arxiv.org/pdf/1904.01502.pdf) \n", "\n", "![title](Mirmin-Peres-Game/Circuit_Operations.png)\n", "\n", @@ -592,7 +587,7 @@ "\n", "$$ Quantum Solution MPG = Initial Vector * Dynamic Matrix(row, column)$$\n", "\n", - "Starting with the first part of the equation, the intial vector, of the MPG circuit is defined by quantum gates as follows:\n" + "Starting with the first part of the equation, the initial vector, of the MPG circuit is defined by quantum gates as follows:\n" ], "metadata": {} }, @@ -613,8 +608,8 @@ "text/plain": [ "
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" }, "metadata": {}, "execution_count": 11 @@ -646,7 +641,7 @@ { "cell_type": "markdown", "source": [ - "In terms of the gates, these are the mathmatical represetations of the relevant gates illustrated above. \n", + "In terms of the gates, these are the mathematical representations of the relevant gates illustrated above. \n", "\n", "$$\n", "I = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} \n", @@ -668,8 +663,7 @@ "\\end{pmatrix}\n", "$$\n", "
\n", - "The \"i\" in CNOT(i, i+2) refers to the control qubit in the first instance and to the target qubit in the second instance under the condition that there are at least two qubits in between such as $CNOT_(1,3)$\n", - "\n" + "The \"i\" in CNOT(i, i+2) refers to the control qubit in the first instance and to the target qubit in the second instance under the condition that there are at least two qubits in between, such as $CNOT_(1,3)$" ], "metadata": {} }, @@ -731,7 +725,7 @@ { "cell_type": "markdown", "source": [ - "This resulting vector holds the relevant qubit states that can be further considered and modified in the dynmaic part of the circuit so that valid winning condition can be created, these Qubit states are:\n", + "This resulting vector holds the relevant qubit states that can be further considered and modified in the dynamic part of the circuit, so that a valid winning condition can be created, these Qubit states are:\n", "$$= \\frac 1 {2} (|0000\\rangle+|0101\\rangle+|1010\\rangle+|1111\\rangle)$$\n" ], "metadata": {} @@ -739,7 +733,7 @@ { "cell_type": "markdown", "source": [ - "Furthermore, when analyzing the initial vector, it becomes apparent that when multiplying any matrix by it, only four relevenate cells per row in the multiplier remain open for consideration, since the others are all multiplied by 0 and thereby likewise drop of with 0% probability as a result. Thus, we can eliminate the majority of the entries whatsoever. As an example we have prepared the matrix below, and denoted all entries with an i, that have to be computed and all others that result in any case with 0 with -.\n", + "Furthermore, when analyzing the initial vector, it becomes apparent that when multiplying any matrix by it, only four relevant cells per row in the multiplier remain open for consideration, since the others are all multiplied by 0 and thereby likewise drop of with 0% probability as a result. Thus, we can eliminate the majority of the entries whatsoever. As an example we have prepared the matrix below, and denoted all entries with an i, that have to be computed and all others that result in any case with 0 with -.\n", "\n", "$$ = \n", "\\begin{bmatrix}\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\ \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} & - & - & - & - & \\color{red}{X} \\\\\\end{bmatrix} \n", @@ -747,7 +741,7 @@ "\\begin{pmatrix} \\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\frac 1 {2} \\\\0 \\\\0 \\\\0 \\\\0 \\\\ \\frac 1 {2}\\end{pmatrix}\n", "$$\n", "\n", - "Let us now take a look at an concrete example, which shows and proofs that only the Xth-columns are relevant for the subsequent computation. Since all other columns are multiplied by zero and thus remain zero:\n", + "Let us now take a look at a concrete example, which shows and proofs that only the Xth-columns are relevant for the subsequent computation. Since all other columns are multiplied by zero and thus remain zero:\n", "\n", "$$\n", "Winnning Plays= \n", @@ -771,14 +765,14 @@ "$$\n", "= \\frac 1 {\\sqrt{8}}\n", "$$\n", - "And indeed, this idea is very powerful because it lets us visually see what outcomes may emerge as well asa better idea of which outcomes may fall off and which may be a win in the MPG. As long as postive result emerges the MPG is won, no matter the probability of that play occuring" + "And indeed, this idea is very powerful because it lets us visually see what outcomes may emerge, as well as a better idea of which outcomes may fall off and which may be a win in the MPG. As long as a positive result emerges, the MPG is won, no matter the probability of that play occurring." ], "metadata": {} }, { "cell_type": "markdown", "source": [ - "Now, in order to consider the later, dynamic, part of the equation we will have to select an example. In this case we have selected row: 1 and column: 1. The corresponding quantum circuit is as follows:" + "Now, in order to consider the latter, dynamic, part of the equation we will have to select an example. In this case, we have selected row: 1 and column: 1. The corresponding quantum circuit is as follows:" ], "metadata": {} }, @@ -801,8 +795,8 @@ "text/plain": [ "
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", 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" }, "metadata": {}, "execution_count": 14 @@ -813,7 +807,7 @@ { "cell_type": "markdown", "source": [ - "The translation of these circuit gate operations is in turn equivalent to the following mathematical matrice multiplications:\n", + "The translation of these circuit gate operations is in turn equivalent to the following mathematical matrices multiplications:\n", "$$ DynamicMatrix(1, 1) = H ⨂ H ⨂ I ⨂ H $$\n", "$$\n", "= \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} ⨂ \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ⨂ \\frac 1 {\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix}\n", @@ -857,7 +851,7 @@ { "cell_type": "markdown", "source": [ - "Now, as we have both, the inital vector and the dynamic matrix, we can compute the winning plays for Alice and Bob with selected column and row of 1. In prinicple we can apply the rule of the i-rows, that we have established above, to this matrix-vector multiplication, however, in this case for the sake of completness we let all entries remain. But feel free to check the rule!" + "Now, as we have both, the initial vector and the dynamic matrix, we can compute the winning plays for Alice and Bob with selected column and row of 1. In principle, we can apply the rule of the i-rows, that we have established above, to this matrix-vector multiplication, however, in this case for the sake of completeness we let all entries remain. But feel free to check the rule!" ], "metadata": {} }, @@ -962,8 +956,8 @@ "text/plain": [ "
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", 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OHDhg8N6ILdXqwF+0aBHNmjWzmtazZ0/8/PxISkoiISEBgLVr15KVlcXKlSstn2QjIyNJT09n2bJlpg781TPCaRUYyb2Pv1yp6XJzcvPh71sh4ydwAgqBvEI4eBoOnIK+QfBQZ4OLtFN5k/+I090huDw9xOhSar2HHnqITz75hAYNGnDo0CHGjx/Ppk2bSrT77LPPmDVrFn5+fkyfPp2nn36at956C8AS9tu2bavp8qWambZLv6CggDlz5uDv74+rqyvBwcGkpqbSvn17Ro0aBVAi7AHuuusuvLy8OHnypGXatWvXAPDw8LBq6+HhQUFBQfXthNQa/0gvCnsoCvtixV376/fCwVM1XVXV6R7rjqtLly6WsH/33Xe5++67Sw376x07doxhw4axdetWy7SVK1cq7O2UaQP/ueeeY/r06cTGxrJ+/XoGDx7MkCFDOHr0KKGhoWUut3//fs6dO0dgYKBl2pAhQ/D09CQuLo7MzEyysrJITk5mw4YNvPDCCzWxO2JiV3Nh+5Hy2zgBqYdrpJxboi5Yx1SvXj2Sk5Np0KAB7733Hn/4wx+4evVqhcsVn7MPDw/nypUrFBQUMHTo0HL/xkrtZcou/RUrVpCcnMzWrVsJCwsDICIigm+++YaPP/64zDdjbm4usbGxeHl5WXoBAHx8fNi+fTuPPvqo5bx+nTp1SExMZOTIkVbrmDZtGqtWreLw4cOsXLmy0s8Xd3JyqlS7QfFbaBEQXqm2tpaaupXxfSJqdJsTlhcdIlf252OEVoGR/H7qF+W2KQQOnynA2bkuhYXG9ApNnDixwjaJiYkVtktMTLRVSRYub87GOTjI5uutjK1bt+J0T/ca294fZ78DFL2nr39tpDFjxhAYGEhGRkaln6dw/QC94m78xx57jEmTJrFw4UK6d7f+maamphq+n1K6yg4+N2Xgv/766zz88MOWsC/Wtm1b6tatS+fOJU+mFhYWEhMTw65du0hJScHT09MyLzMzkwEDBuDj40NCQgLu7u5s3ryZSZMmkZ+fz4QJEyxt/f39mT9/Pq+88kr17WAV7Vo7k6/XzbGalnvlMq0CIw2qyL4416lXqXZOTs44u9QhP+9aNVckUnlOTk6WpyROnjyZnJycCpcpLey3bdvGnj17GDFiBN26daNr167s2bOnusuXGmS6wD958iT79+8v9SjlxIkTdOrUifr165eYN3bsWJYvX86KFSvo06eP1bypU6eSnZ1NSkoKrq6uAISHh5OXl8eUKVN45plnLB8QnnnmGQBmzpxZpbor+wlrwRfww9kqrZp7H4svddBeVYWFhbN6Rs1eb/b8B0X/mvnyx58vwYzPKm7X9DbIy624m7S6HD5c8TmFxMREq96t0sydO9dWJVlE7tnJvy781+brrYzw8HA21eD7a2rC34Ci9/T1r2tKeHi41ViNHj160KZNG44fP05KSkqFy5cV9gA5OTksWbKEyZMnM3ToUKvADwsLszrXL7WP6c7hFw+2a968udX0nJwcUlNTS+3OnzRpEklJSbz77rtER0eXmJ+WlkZgYKAl7It17dqVa9eu8f3339twD6S28WoE7ZsXnacvT8/2NVLOLXnttdeMLkFq2L333gvAxo0bKxyEXF7YF1u/fj1QdHmf2BfTBb6XlxcAGRkZVtPfeOMNzpw5Q5cuXaymx8fHM3fuXBYsWMCzzz5b6jq9vb3Zt29fia6u7du3A0Xn+MWx/b4r1K9bdui3bAr3+ddoSTdl8ODBRpcgNaz4FGdaWlq57SoT9tevJyjImDEZUn1M16XfunVrgoKCmDVrFk2bNsXHx4fVq1ezbt06AKsj/DfffJNZs2YRHR1NaGgoO3bssMxzd3e3PEhk3LhxDBo0iL59+zJ+/Hjc3d3ZtGkT8+fPJyoqipYtW9bsTorpNG8ME/rAmt1w5LpTLs5O0NUPBnaF+qb7bSkpICCAQ4cOGV2G1KAvv/ySwsLCCgN/6tSpFYY9wIULF1i6dGmlxgJI7WK6P2HOzs6sWrWK2NhYRo8ejaenJ8OHD2fMmDHEx8dbfeosPl/10Ucf8dFHH1mt5/rzTQMHDmTjxo3Mnj2b0aNHk52djZ+fHzNnzrQasGdGUS9vrdJ0uXneHjD2QfhPFrz+j6JpfxkIbq7lLia3qM6cBKNLqNXee+893nvvvQrbLVy4kMjISKZNm1bhdfZl9ZZK7Wa6wAdo164dW7ZssZo2dOhQAgICaNCggWVaVQaQ9OnTp8RgvtLk5uaSn59PQUEBubm5XLlyhfr16+tyFAdyR+PfXivsxV5kZWXRu3dvo8sQA5nuHH5Z9uzZUyM3g/jDH/5AgwYN2LZtG0899RQNGjQgMzOz2rcrYgvh4eFGlyAiJlUrAv/y5ctkZGSUGLBXHZYuXUphYaHVV/HNekTMLikpyegSRMSkTNmlfyM3Nzc9k1mkEkaPHq3QF5FS1YojfBGpHN0YRUTKosAXERFxAAp8ERERB6DAF7EjuumOiJRFgS9iR1auXGl0CSJiUrVilL498WnimNuWmjFt2jRD7qcf3KhRjW/TDNs2QkhISJWXOXriDACtW3lbva7u7Yq5KPBr2MCuRlcgYnt/bd/R6BIcxrx586q8TPFjfGf/cZTVa3Es6tIXERFxAAp8ETuyaNEio0sQEZNS4IvYkU6dOhldgoiYlAJfxI6EhYUZXYKImJQCX0RExAEo8EVERByAAl/Ejtxzzz1GlyAiJqXAF7Eju3fvNroEETEpBb6IiIgDUOCLiIg4AAW+iB1ZvXq10SWIiEkp8EVERByAAl/EjkRFRRldgoiYlJ6WJyK37OM9cOqCMdv2aaKnUJrd888/T3p6uiHbDgkJuaknDNojBb6I3LJTF+CHs0ZXIWaVnp5Oamqq0WU4PHXpi9iRMWPGGF2CiJiUAl/EjowdO9boEkTEpBT4YnNXcuHbE/CP9N+mLd0GG/fBodOQX2BYaXavV69eRpcgIialc/hiMxdz4PP9sPsoXM2znpd+ougLwN0VevhD745QX+9Amzp37pzRJYiISenPrdjE18dh9W7IuVZx24tXio729xyDp7pDmzuqvTwREYenLn25ZZ/vg/e/rFzYX+/8ZVi4GdIzq6cuR9SxY0ejSxARk1Lgyy356ntYt/fmly8ohGVfwvf/sV1NjmzNmjVGlyBiE05OTvj7+9O7d28iIyMJCQmhXr16ZbavX78+EydOxMXFpQarrF1qdeBv3ryZ4cOH4+/vT8OGDfH19WXYsGEcO3asRNu0tDQGDBhA8+bNcXNzIzg4mKSkJAoLCw2o3D78fAk++br8NvOeLvoqT0EhrNheNNhPbs2rr75qdAkiN83JyYnIyEjWrFnDL7/8QkZGBps3b+aLL74gLS2NS5cusW3bNoYNG4arq6tlufr167NmzRrmzp2rm+yUo1YH/ttvv83p06eZMmUK69evZ8aMGezcuZPQ0FCOHz9uaZeZmUlERASnT59m4cKFfPrpp/Tq1Yu4uDjmzJlj3A7Ucp98Dbn5tlnXhWzYfMA263Jkq1atMroEkZvSvn17vvzyS7744gsGDhyIu7s7J0+eZMuWLWzatInDhw9Tp04d7r//fpKTk/n+++956KGHLGHfv39/zp07x9/+9jejd8W0avWgvUWLFtGsWTOraT179sTPz4+kpCQSEhIAWLt2LVlZWaxcuZI2bdoAEBkZSXp6OsuWLePFF1+s8dpru/OX4eAp267zqyPwUGeoox45u7d6RjitAiO59/GXKzVd7Ft0dDRLly7F1dWVM2fOsHDhQpKTkzl58qRVO3d3dwYNGsSECRMIDg5mw4YNHD16lNatW3Pu3DkeeOAB9u3bZ9BemJ9pj/ALCgqYM2cO/v7+uLq6EhwcTGpqKu3bt2fUqFEAJcIe4K677sLLy8vqjXLtWtFoMg8PD6u2Hh4eFBToovCb8fVxsPXJkOyrRdfpi4jjGDx4MCtWrMDV1ZWlS5cSEBDAzJkzS4Q9wMWLF1myZAmhoaH86U9/oqCggNatW5OTk6OwrwTTBv5zzz3H9OnTiY2NZf369QwePJghQ4Zw9OhRQkNDy1xu//79nDt3jsDAQMu0IUOG4OnpSVxcHJmZmWRlZZGcnMyGDRt44YUXamJ37M6J87VrvY5C9yuX2qRNmzYsWbIEZ2dnXnnlFZ599lmysrIqXK64a9/Z2ZnCwkIaNGhAhw4daqDi2s2Ugb9ixQqSk5P57LPPmDx5MhEREcTHx9OjRw/y8vLKDPzc3FxiY2Px8vKy9AIA+Pj4sH37dvbu3Yuvry8eHh7ExMSQmJjIyJEjLe2uXr3KiBEj8PHxwcPDg969e3Po0KFq39/a6HQ1PRnt9C/Vs15HceCABkJI7bF48WIaNmzI8uXLmTFjRqWWufGc/fTp04GiU7yenp7VWW6tZ8pz+K+//joPP/wwYWFhVtPbtm1L3bp16dy5c4llCgsLiYmJYdeuXaSkpFj9x2dmZjJgwAB8fHxISEjA3d2dzZs3M2nSJPLz85kwYQIAeXl5tG3blpkzZ9K8eXMSEhKIjo5m796KrztzcnK6xb2uXf6w6CwN3a1PqZQ3Gr+sec9/YP39xk1bGRURcYvV3boJy4tOWJjp/3XixIkVtklMTKywXWJioq1KshgUv4UWAeFVWmbX2pl8vc560Gzulcu0Coys0npSU7cyvk/NvWf+OPsdoOi9cf1rMzNjzffddx9hYWH8/PPPjB8/vlLL3Bj2xd34PXv2JCIigpiYGMvYrWKpqamG72t1q+zVZqYL/JMnT7J///5S/2idOHGCTp06Ub9+/RLzxo4dy/Lly1mxYgV9+vSxmjd16lSys7NJSUmxXMoRHh5OXl4eU6ZM4ZlnnsHT05PbbruNl1/+bbDQuHHjiI+P58qVK1aXgAgU5FXxLjuVlJ97tVrWK+Zz72PxpQ7aE8cwevRoAN555x0uXKi4y7CssAdISEggIiKC//mf/+GNN97Q5dZlMGXgAzRv3txqek5ODqmpqfTr16/EMpMmTSIpKYnFixcTHR1dYn5aWhqBgYElQrtr165cu3aN77//vtSuoK+++gpfX99Khb2jvcEWbip5s5wbj9bhtyP70uaV5pknHuKTBON/lsX1mun/9fDhwxW2SUxMtDqdVZq5c+faqiSLBV/AD2dtvtpKCQsLZ/WMmvt/mppQdNlXYWGh1WszM7rm8PDwEuNLHnjgAQDef//9CpcvL+wBPv/8c3766Sd8fX1p3bo1P/zwg2VeWFgYW7dutc2O1HKmO4fv5eUFQEZGhtX0N954gzNnztClSxer6fHx8cydO5cFCxbw7LPPlrpOb29v9u3bR05OjtX07du3A0Xn+G904cIFxowZw8yZM296X+xZi6bVs96W1bReR/Haa68ZXYJIhZo3b463t7fl5jrlqSjsoejDy549ewDKHdTt6Ex3hN+6dWuCgoKYNWsWTZs2xcfHh9WrV7Nu3TrA+j/zzTffZNasWURHRxMaGsqOHTss89zd3S33FR83bhyDBg2ib9++jB8/Hnd3dzZt2sT8+fOJioqiZcuWVjXk5OTw6KOPEh0dzVNPPVUDe137BLWELTYez1jHGTreadt1OprBgwcbXYJIhYoPso4ePVpuT0Nlwr7YkSNHALjzTv0RKYvpAt/Z2ZlVq1YRGxvL6NGj8fT0ZPjw4YwZM4b4+HiCgoIsbVNSUgD46KOP+Oijj6zWc303zsCBA9m4cSOzZ89m9OjRZGdn4+fnx8yZMy0D9orl5eUxePBg/P39dXRfDl8v8GkCp2w4Wj+kFbhpqMQtCQgIqBVXlkS9vLVK08W+fPPNNzRq1Ig6dcqPoPr163P77bdX6qY6L7/8MtOmTSM7O9vW5doN0wU+QLt27diyZYvVtKFDhxIQEECDBg0s06pyXqZPnz4lBvOVJiYmhoKCAt2esQJOTvDo3ZD0T9usr64LPBxUcTsRqf0KCwu5fPlyhe0uXrzIgw8+yJ133lnhB1kFfcVMGfil2bNnD927d6/WbWRmZpKcnIyrq6vVXfkOHjxIq1atqnXbtVF7b7jPv+iJebfq0bvBq9Gtr0dE7EtWVlalbsYjFasVgX/58mUyMjKIi4ur1u3cddddph9tazYDQ4sefFPWLXErMzq/V3u4v51t63JU4eHhRpcgIiZVKwLfzc2N/HwbPZZNbKqOC4zsBWv2wPYjVVvW2Qn6BkFkp6JTBHLrkpKSjC5BREzKdJflSe1TxwWiu0FsBDRvXLll2twOk/rCg4EKe1sqvpmJiMiNasURvtQOAXdCB284ehbST8CP/4X/XIS8fKhfB7w9oJUnhPrCnU2MrtY+6QYjIlIWBb7YlJMTtLmj6EtERMxDXfoiIiIOQIEvYkdqw013RMQY6tIXsSMrV6405Pa6PgaOyTBy21I5ISEhN7Xc0RNnAGjdytvqdU1s2x4p8EXsyLRp0wwJ/IFda3yTUovMmzfvppYrfrLf7D+OsnotN0dd+iIiIg5AgS8iIuIAFPgidmTRokVGlyAiJqXAF7EjnTp1MroEETEpBb6IHQkLCzO6BBExKQW+iIiIA1Dgi9iRe+65x+gSRMSkFPgidmT37t1GlyAiJqXAFxERcQAKfBEREQegwBexI6tXrza6BBExKQW+iIiIA1Dgi9iRqKgoo0sQEZPS0/JE5JZ9vAdOXTBm2z5N9LQ+qR7PP/886enpNb7dkJCQm37CYHkU+CJyy05dgB/OGl2FiG2lp6eTmppqdBk2oy59ETsyZswYo0sQEZNS4IvYkbFjxxpdgoiYlAJf5Dr5BXD6unPR5y9DYaFx9VRVr169jC5BRExK5/DF4eXlQ/oJ2H4EMs8XfV9s+lpoWA/ae8P9/tD6dnByMq7Wipw7d87oEkTEpBT44tAOn4H/2wG//Fp2m1+vQVpm0Veb22FId/BqVHM1iojYgrr0xSEVFsJn38Db/yw/7G/0w1lISIG9P1ZfbbeiY8eORpcgIialwBeHU1gIn3wN/zx0c8vn5sOSf5kz9NesWWN0CSJSBW5ubri4uNTItmp14G/evJnhw4fj7+9Pw4YN8fX1ZdiwYRw7dqxE27S0NAYMGEDz5s1xc3MjODiYpKQkCmvTiCyxibRM+Nd35beZ93TRV1kKgQ++KhrUZyavvvqq0SWIOCRPT09GjRrF3//+d7Zv387evXvZvXs3y5YtY/z48bRs2bLEMu7u7nz++ecsW7asRkK/Vgf+22+/zenTp5kyZQrr169nxowZ7Ny5k9DQUI4fP25pl5mZSUREBKdPn2bhwoV8+umn9OrVi7i4OObMmWPcDkiNu3wF1tjokfFX8+CjnbZZl62sWrXK6BJEHIq3tzdLlizh5MmTvPPOO8TExNC9e3c6d+5M165dGTp0KPPnz+fYsWN88skndOjQASgK+w0bNtCjRw/uu+8+mjVrVu211upBe4sWLSrxQ+rZsyd+fn4kJSWRkJAAwNq1a8nKymLlypW0adMGgMjISNLT01m2bBkvvvhijdcuxth+BLKv2W59GT/BifPQytN263QEq2eE0yowknsff7lS00XM6Mknn2TRokU0adKEgoIC1q1bx8aNG0lLS+OXX37htttuIygoiN69e/P73/+exx9/nIcffpgZM2bQv39/evTowfHjxwkPD+enn36q9npNe4RfUFDAnDlz8Pf3x9XVleDgYFJTU2nfvj2jRo0CKPUT0V133YWXlxcnT560TLt2regvvIeHh1VbDw8PCgoKqm8nxFQKCuHL722/3n9n2H6dImJuEydO5MMPP6RJkyakpKTg7+9P//79eeutt9i2bRv79u1jx44d/O1vf+PJJ5+kZcuWvPvuu7i6ujJjxgyrsM/MzKyRmk0b+M899xzTp08nNjaW9evXM3jwYIYMGcLRo0cJDQ0tc7n9+/dz7tw5AgMDLdOGDBmCp6cncXFxZGZmkpWVRXJyMhs2bOCFF16oid0REzh3qWoj8isro/o/mFeaPd33W8SsnnjiCebOnQvA+PHjeeSRRzh69Gi5y5w9e5ZJkyZx+PBhy7SVK1fWWNiDSQN/xYoVJCcn89lnnzF58mQiIiKIj4+nR48e5OXllRn4ubm5xMbG4uXlZekFAPDx8bEMovD19cXDw4OYmBgSExMZOXKk1Tqefvpp7rjjDho3bsy9997L9u3bq3VfpeacPF896/3l16KxAWZw4MABo0sQsWu33347SUlJQNHT9BYsWFCp5YrP2Xfo0IH//Oc/luU7d+5cbbXeyJTn8F9//XUefvhhwsLCrKa3bduWunXrlvoDKiwsJCYmhl27dpGSkoKn528nVTMzMxkwYAA+Pj4kJCTg7u7O5s2bmTRpEvn5+UyYMMHSNj4+niVLllCvXj1SUlIYNGgQp0+frrBmJzPffk0A6NJ/Mj2HvGk1rbyR+OXNf/4D6+/92ofw84lvb6G6ik2cOLHCNomJiRW2S0xMtFVJFoPit9AiILxKy+xaO5Ov11kPms29cplWgZFVWk9q6lbG94mo0jK34o+z3wGKfuevf21mtbFmMGfdf/7zn/H09OTzzz9n/vz5lVrm+gF6xd34U6ZMIS4ujrlz5/Lggw9atU9NTa3Sflb2ajPTBf7JkyfZv39/qX+0Tpw4QadOnahfv36JeWPHjmX58uWsWLGCPn36WM2bOnUq2dnZpKSk4OrqCkB4eDh5eXlMmTKFZ555xvIBofjGJYWFhdStW5effvqJK1euWJaT2suJ6vtDYfQfodro3sfiSx20J2JW7u7uDB06FCg6Oq/sMjeGfWZmJi+99BLDhw8nMjKS9u3b8913FVwrbAOmDHyA5s2bW03PyckhNTWVfv36lVhm0qRJJCUlsXjxYqKjo0vMT0tLIzAwsERod+3alWvXrvH9999b9Qg8/fTTrFmzhqtXrzJmzJhKhb2u5ze/nT/Ahzusp914pF6s+Mi+rPk3yjiYhkfDm6+tMq4/91eWxMREq9NZpSk+92hLC74ouguhEcLCwlk9o+Z+/6Ym/A0o+p2//rWZ1caawfi6w8PDrcbFPPLII7i5ubF161YOHar4zl1lhT1AVlYWH374ITExMURHR/OXv/zFslxYWBhbt261+f6Y7hy+l5cXABkZ1kOf33jjDc6cOUOXLl2spsfHxzN37lwWLFjAs88+W+o6vb292bdvHzk5OVbTi8/P+/j4WE3/4IMPuHTpEp9++indu3e/pf0R82jRtHrW61YfGjeonnVX1WuvvWZ0CSJ2q2vXrgB88cUXFbYtL+yLFa+neL3VzXRH+K1btyYoKIhZs2bRtGlTfHx8WL16NevWrQOwGrD35ptvMmvWLKKjowkNDWXHjt8O39zd3S3d8+PGjWPQoEH07duX8ePH4+7uzqZNm5g/fz5RUVGl3gGpbt26PPbYYwQHB3PvvffSrl27at5zqW7NGxc9+e5XG16HD+Z6gt7gwYONLkHEbnXq1AmAb78tf7xOZcIeID093Wq91c10ge/s7MyqVauIjY1l9OjReHp6Mnz4cMaMGUN8fDxBQUGWtikpKQB89NFHfPTRR1brub5LZODAgWzcuJHZs2czevRosrOz8fPzY+bMmVYD9kpz7do1jh8/rsC3Ay7O0K0NbLnJe+iX5T5/267vVgQEBFSqq9FoUS9vrdJ0ETP44IMP2L17d4VXwyQkJFTqOvszZ84wa9Ysfv755+ootwTTBT5Au3bt2LJli9W0oUOHEhAQQIMGv/WdVuUcR58+fUoM5rvR+fPn+ec//0n//v2pU6cOf//73zl9+nSJ0whSe93fDrZlWD/z/lbc6QHtmlfYTETswLJlyyrVburUqXh5eTF58uRyr7O/dOkS8fHxtiqvQqY7h1+WPXv2lHvDHVtZsGAB3t7e3HHHHfzf//0fKSkplnEFUvt5ukH/YNusy9kJhvQo+ldEpFhWVhZPPPFEjd5UpzJMeYR/o8uXL5ORkUFcXFy1bsfT05N//etf1boNMV5YezjyHzhwquw2lRmd/2gXaFlNAwFvVnh4uNEliIhJ1YrAd3NzIz/fRn2w4vCcnWFET0j+N+w/WXH70jx6N4R3sG1dtlB8BzARkRvVmi59EVuq6wLP9YKBXYteV5anG4yJhN4dq6+2WzF69GijSxARk6oVR/gi1cHZCXq1h6CW8NX3sOMIXCzjnvh3NoH7/aGrH9Qz8W9NddysQ0Tsg4n/dInUDI+G0C8Y+gbB+ctw8r+QfbXo2nqPhtDSExrpzsoiUssp8EX+f05O4NWo6EtExN7oHL6IHakNN90REWMo8EXsyMqVK40uQURMSl36InZk2rRphtxP36dJjW/SFNsW+xYSElLlZY6eOANA61beVq+re7uVocAXkVs2sGYe9iVSo+bNm1flZYof4zv7j6OsXpuBuvRFREQcgAJfxI4sWrTI6BJExKQU+CJ2pKaeqy0itY8CX8SOhIWFGV2CiJiUAl9ERMQBKPBFREQcgC7LE6klOnSo+Hm806ZNq1Q7EXE8OsIXsSN//vOfjS5BRExKgS8iIuIAFPgiIiIOQIEvIiLiABT4IiIiDkCBLyIi4gAU+CIiIg5AgS8iIuIAFPg1LC4ujjp1dL8jEUe0detWOnXqRNu2bYmJiSE/P9/okio0btw4WrRoUav+bv3444888MADBAQEEBgYyJ/+9CejS6q0Bx98kODgYIKCgoiKiuLixYs2W7cCvwZt27aN7Oxso8sQEQMUFBQQExPDqlWrOHLkCBcvXmT58uVGl1Wh6Ohovv76a6PLqJI6deqQkJDAoUOH+Oabb/jqq69Yu3at0WVVyurVq/n222/Zu3cvrVq1Yu7cuTZbtwK/hly9epWpU6cyZ84co0sREQPs3r2bO++8k44dOwIwcuRI1qxZY3BVFbv//vu54447jC6jSry9venatSsA9erVIygoiBMnThhcVeU0btwYKPqAmJ2djZOTk83W7VRYWFhos7VJmeLj42nTpg3PPfccderUIS8vz+iSRKQS9h4+ypbtaZbvz5w9D4D37Z5WrwGaejTiqccicXEueSy1Zs0aPv74Yz744AMADh06xFNPPUVaWlqJtrfq2rVckj/eyK85VyusGWBA5H20buld7jpr4u/W9m8OsOvbw5bvy6vbp7kXgx7uVWEgnj9/npCQEL744otqec7E5V9zWLZmI7l5+RXWDPDkIxHc0axpuet89NFH2bFjBx07duQf//gHbm5uNqlVR/g1YO/evezcuZNnn33W6FJEpIoC2/lSr24dzpw9b/kDDpR4/dO5/9LznqBSwx6gsLDQKpyq81irXr26dAvpWGHNZ86ex6uJO34tmldbLVUR2rk9eXl5FdZ97r+/0Ove4ArD/urVq0RFRfH8889X20Ol3Bo2IDigbaV+1n4tm1cY9gCfffYZP/30Ez169OCdd96xWa0K/Brw5ZdfcvDgQfz8/PD19SU/Px9fX1+bDsYQkerh7OzM4P4R1Ktb/qC1sG7B+JYTnC1btrTqVv7xxx9p0aKFzeq8UVCH1tzdqW25bRrd1oDHH+pp027jW1Gvbh0GPxKBcwX19A3vxu2eHuW2yc/P5+mnn+buu+9m0qRJNqyypB6hnWh7l0+5bZo1bczDYd0qvU5nZ2dGjBjB0qVLb7G669ZpszVJmUaPHs3p06c5fvw4x48fx8XFhePHj+Pu7m50aSJSCZ5N3Hmkd48y53vf7knk/aHlrqNr166cOnWKgwcPArB48WIGDhxo0zpv9Gjk72jc6LYy50f1DeO2Bq7VWkNVtfS+nd73dSlzftu7fOjRpVOF6xk1ahSNGjXir3/9qy3LK5WzkxNP9AvDtX690uc7OxH9SO8KPzRevHiRM2fOWL5fs2YNnTpVvK+VrtNma6rFwsPDcXJyKvWrup8triEUIrXDPcEd6NCmVYnpdVxciH4kgjouLuUu7+Liwt///neioqJo06YNbm5uDB06tLrKBaCBa32e6B9e6rxuIQG0L2V/bhQbG0uLFi3Iz8+nRYsWjBkzxsZVlhTR425aejcrMd21fj2e6BdWYQ/Al19+yXvvvceePXu4++67CQkJ4a233qqucgFo7O7G433uL3Ve7/u60KKU/blRVlYWjz76KJ07dyYoKIj09HTmz59vsxo1aA84ePBgie717777jhEjRjBlyhQSEhKqbdvrt+7kcnYOgyrxJhYRY126/Cvz3ltNds4Vy7T+Ed3peW+QgVVV7P9t/oov9+y3fO/ZxJ0JIwZRr15dA6sq37n//sJbS9ZYBsMBPDmgNyEdyz9NYbQVazex9/BRy/ctvZvxP888VubYjppkfAUm0LFjR7p37271dejQIQBGjBhRbdu9nJ3DV1/vp6CwQGEvUgs0cmvI7x/uafm+das7+d09nQ2sqHIe7nUvt3s2AcDJyYno/hGmDnuAZk096BfR3fJ9cEAb04c9wON97sfdrSEAdeu4MPiRCFOEPThQ4O/bt49Bgwbh5eWFq6sr/v7+xMfHl9o2Pz+f999/n27duhEQEFBtNf1r17fk5RfQu0fZ56tExFwC2/nRJbAd9evVZXD/8FrxYb1u3TpED4jA2dmJiO4htPKpHdfVd7+7I+38WuDu1pDHHvyd0eVUSsMGrkT1CwegX0R3mjX1MLSe6zlEl/7XX39Nr169uOuuu5gyZQqtWrXi2LFjfPXVVyxevLhE+/Xr19OvXz/efvttYmNjK7WNqQl/s3XZIiIiFZr9x1GVald7bo58CyZNmkSjRo3YsWOH1cj4kSNHltp+6dKluLq6Eh0dXVMlioiIVCu7P8L/9ddfadSoEePGjWPevHkVtr9w4QLe3t4MHDiQFStWVEtNl7NzSHh7BYHt/Yh+pHe1bENEROR6dn+Ef+HCBQoKCvDxKf+mCMU+/PBDrl69WuXBejfTpZ924AhpB45UeTkREZFile3St/tBe02aNMHZ2ZlTp05Vqv2SJUto0aIFkZGR1VyZiIhIzbH7Ln2AiIgIDh06REZGRrl3tztw4ACBgYG89NJLzJw5s1pqWbdlB9t27+OFkU/QrIJbQ4qIiNiK3R/hA8yZM4dLly7RvXt3li5dypYtW0hOTiYmJsaq3ZIlS4Dqu/b+cnYO2785QEjHNgp7ERGpUXZ/Dh8gNDSU7du388orrzBx4kSuXLlCy5YtefLJJy1t8vLy+OCDD/jd736Hv79/tdRx/kIWbrc11HX3IiJS4xyiS99MCgoKcDbJXZdERMRxKPBFREQcgA41RUREHIACX0RExAEo8EVERByAAl9ERMQBKPBFREQcgAJfRETEASjwRUREHIACX0RExAEo8EVERByAAl9ERMQBKPBFREQcgAJfRETEASjwRUREHIACX0RExAEo8EVERByAAl9ERMQBKPBFREQcgAJfRETEASjwRUREHIACX0RExAEo8EVERByAAl9ERMQBKPBFREQcgAJfRETEASjwRUREHIACX0RExAEo8EVERBzA/wfRp1aeF5GjDwAAAABJRU5ErkJggg==" }, "metadata": {}, "execution_count": 16 @@ -1225,11 +1219,11 @@ "1. Install additional libararies:
\n", "\"!pip install \"git+https://github.com/qiskit-community/qiskit-textbook.git#subdirectory=qiskit-textbook-src\"\"\n", "\n", - "2. Imports libararies:
\n", + "2. Imports libraries:
\n", "from qiskit_textbook.tools import array_to_latex;\n", "from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister, BasicAer;\n", "\n", - "3. Intialise a n-Qubit system and apply gates, an example 2-Qubit system:
\n", + "3. Initialize an n-Qubit system and apply gates, an example 2-Qubit system:
\n", "qc = QuantumCircuit(QuantumRegister(2) , ClassicalRegister(2))
\n", "result = execute(qc, BasicAer.get_backend('unitary_simulator')).result().get_unitary(qc)
\n", "array_to_latex(result, pretext=\"\\\\text{Circuit = }\\n\") \n", @@ -1247,9 +1241,9 @@ { "cell_type": "markdown", "source": [ - "Thus far all executions of the Quantum circuits were performed on ideal quantum simulators, rather than actual quantum hardware. Real quantum computers have noise, decoherence and other periphery hardware issues, which can reduce the performance and fidelity of the executions. Hence, in order to ascertain the actucal performance and not only the theoretical we will execute our circuit on real quantum computers as well as simulators of real computers that include the above mentioned errors alike. \n", + "Thus far, all executions of the Quantum circuits were performed on ideal quantum simulators, rather than actual quantum hardware. Real quantum computers have noise, decoherence and other periphery hardware issues, which can reduce the performance and fidelity of the executions. Hence, in order to ascertain the actual performance and not only the theoretical, we will execute our circuit on real quantum computers as well as simulators of real computers that include the above-mentioned errors alike. \n", "\n", - "We start off with the non-ideal simulators of the real quantum computers, so called FakeDevices. The use of FakeVigo, simualating the quantum Computer Vigo yields the following results." + "We start off with the non-ideal simulators of the real quantum computers, so called FakeDevices. The use of FakeVigo, simulating the quantum Computer Vigo, yields the following results:" ], "metadata": {} }, @@ -1622,7 +1616,7 @@ { "cell_type": "markdown", "source": [ - "We can oberserve that simulated noisy devices outperform classical systems and thus indicate that real quantum computers are capable of doing the same. Now in order to evaluate the performance of real devices we select the ibmq_athens computer to benchmark the MPG. Please note that you will have to add your own API token from the IBM Quantum website (https://quantum-computing.ibm.com) so that this cell executes" + "We can observe that simulated noisy devices outperform classical systems, and thus indicate that real quantum computers are capable of doing the same. Now, in order to evaluate the performance of real devices, we select the ibmq_athens computer to benchmark the MPG. Please note that you will have to add your own API token from the IBM Quantum website (https://quantum-computing.ibm.com) so that this cell executes" ], "metadata": {} }, @@ -2088,7 +2082,7 @@ { "cell_type": "markdown", "source": [ - "As outlined above, the MPG circuit outperforms the classical circuit by about 7.8% (88.89% vs. 96.69%) on a real, noisy, quantum computer. Further optimising can be achieved via measurement error mitigation as can be further seen in this [notebook](https://qiskit.org/textbook/ch-quantum-hardware/measurement-error-mitigation.html)." + "As outlined above, the MPG circuit outperforms the classical circuit by about 7.8% (88.89% vs. 96.69%) on a real, noisy, quantum computer. Further, optimizing can be achieved via measurement error mitigation, as can be further seen in this [notebook](https://qiskit.org/textbook/ch-quantum-hardware/measurement-error-mitigation.html)." ], "metadata": {} }, @@ -2834,8 +2828,8 @@ "text/plain": [ "
" ], - "image/png": 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", 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" }, "metadata": {}, "execution_count": 32 @@ -2846,7 +2840,7 @@ { "cell_type": "markdown", "source": [ - "The noisy could be reduced in this experiment by further 4%, however, the orginal results could not be reconstricuted and thus attained or surpassed. At this point, if running this notebook on a shared system (e.g. mybinder.org). Remember to log off from your IBM Q account in case you used it for execution on real devices by running the following cell:" + "The noisy could be reduced in this experiment by further 4%, however, the original results could not be reconstructed and thus attained or surpassed. At this point, if running this notebook on a shared system (e.g. mybinder.org). Remember to log off from your IBM Q account in case you used it for execution on real devices by running the following cell:" ], "metadata": {} }, @@ -2863,7 +2857,7 @@ { "cell_type": "markdown", "source": [ - "#### Finally, this conclues our experiments on the Mermin Peres Magic Sqaure Game. We elaborated the quantum architecure of the game, presented its scheme of operation and quantified the results on real quantum computers. \n", + "#### Finally, this concludes our experiments on the Mermin-Peres Magic Square Game. We elaborated the quantum architecture of the game, presented its scheme of operation and quantified the results on real quantum computers. \n", "\n", "_Dr. Jan-Rainer Lahmann - David Drexlin_" ], @@ -2872,8 +2866,8 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.9.1 64-bit ('quan': conda)", - "name": "python391jvsc74a57bd0475b625634d962d9a94dcb56ed23109cae51528bdc211180c80ccb0ab7f54442" + "name": "python3", + "display_name": "Python 3.9.1 64-bit ('quan': conda)" }, "language_info": { "codemirror_mode": { @@ -2886,6 +2880,9 @@ "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.1" + }, + "interpreter": { + "hash": "475b625634d962d9a94dcb56ed23109cae51528bdc211180c80ccb0ab7f54442" } }, "nbformat": 4, From f01a00808f551b9f000a8f066bce7af4b76e7639 Mon Sep 17 00:00:00 2001 From: David Jacob Drexlin Date: Wed, 25 Aug 2021 19:20:32 +0200 Subject: [PATCH 29/30] feat: powerpoint presentation --- ...Square_Game_on_Real_Quantum_Computers.pptx | Bin 0 -> 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