Be a bit careful about complexity of the sparse Hamiltonian (#1512)

Be a bit careful about complexity

Co-authored-by: Matthew Harrigan <mpharrigan@google.com>

Author: fdmalone

qualtran/bloqs/chemistry/resource_estimation.ipynb

@@ -64,12 +64,12 @@
     "\n",
     "| Name | $$V_{pqrs} $$ | Space complexity |\n",
     "| :---- | :---- | :---- | \n",
-    "| [Sparse](sparse/sparse.ipynb)   | $$\\tilde{V}_{pqrs}$$ | $$O(N^3)$$ |\n",
+    "| [Sparse](sparse/sparse.ipynb)   | $$\\tilde{V}_{pqrs}$$ | $$O(N^4)$$ |\n",
     "| [Single Factorization](sf/single_factorization.ipynb) | $$\\sum_X^{L} L_{pq}^X L_{rs}^X $$ | $$O(N^3)$$ |\n",
     "| [Double Factorization](df/double_factorization.ipynb) | $$\\sum_X^{} \\left(\\sum_k^{\\Xi} U^{X}_{pk}f_k^X U_{qk}^{X*}\\right)^2$$ | $$O(N^2\\Xi)$$ |\n",
     "| [Tensor Hypercontraction](thc/thc.ipynb) | $$\\sum_{\\mu\\nu}^{M} \\chi_p^{\\mu}\\chi_q^{\\mu}\\zeta_{\\mu\\nu}\\chi_r^\\nu\\chi_s^\\nu$$ | $$O(N^2)$$ |\n",
     "\n",
-    "where the space complexity column represents the amount of classical data ($\\Gamma$, say) required to specify the Hamiltonian. The Toffoli complexity for state preparation for all of these approaches goes roughly like $\\sqrt{\\Gamma}$. "
+    "the space complexity column represents the amount of classical data ($\\Gamma$, say) required to specify the Hamiltonian. The Toffoli complexity for state preparation for all of these approaches goes roughly like $\\sqrt{\\Gamma}$. Note under certain conditions sparsity may lead to $O(N^3)$ non-zero elements. "
    ]
   },
   {