Update classical_shadows.ipynb

Author: PeilinZHENG

docs/source/tutorials/classical_shadows.ipynb

@@ -30,7 +30,7 @@
   {
    "cell_type": "markdown",
    "source": [
-    "Let's first briefly review the classic shadows in Pauli basis. For an $n$-qubit quantum state $\\rho$, we randomly perform Pauli projection measurement on each qubit and obtain a snapshot like $\\{1,-1,-1,1\\cdots1,-1\\}$. This process is equivalent to apply a random unitary $U$ to $\\rho$ and measure in computational basis to obtain $|b\\rangle=|s_1\\cdots s_n\\rangle,\\ s_j\\in\\{0,1\\}$:\n",
+    "Let's first briefly review the classic shadows in Pauli basis. For an $n$-qubit quantum state $\\rho$, we randomly perform Pauli projection measurement on each qubit and obtain a snapshot like $\\{1,-1,-1,1,\\cdots,1,-1\\}$. This process is equivalent to apply a random unitary $U$ to $\\rho$ and measure in computational basis to obtain $|b\\rangle=|s_1\\cdots s_n\\rangle,\\ s_j\\in\\{0,1\\}$:\n",
     "$$\n",
     "\\begin{equation}\n",
     "    \\rho\\rightarrow U\\rho U^{\\dagger}\\xrightarrow{measure}|b\\rangle\\langle b|,\n",
@@ -54,7 +54,7 @@
     "    \\rho=\\mathbb{E}\\left[\\mathcal{M}^{-1}(U^{\\dagger}|b\\rangle\\langle b|U)\\right].\n",
     "\\end{equation}\n",
     "$$\n",
-    "We call each $\\rho_i=\\mathcal{M}^{-1}(U_i^{\\dagger}|b_i\\rangle\\langle b_i|U_i)$ a shadow snapshot state and their ensemble $S(\\rho;N)=\\{\\rho_i|i=1\\cdots N\\}$ classical shadows."
+    "We call each $\\rho_i=\\mathcal{M}^{-1}(U_i^{\\dagger}|b_i\\rangle\\langle b_i|U_i)$ a shadow snapshot state and their ensemble $S(\\rho;N)=\\{\\rho_i|i=1,\\cdots,N\\}$ classical shadows."
    ],
    "metadata": {
     "collapsed": false
@@ -70,7 +70,7 @@
     "    \\rho&=&\\frac{1}{N}\\sum_{i=1}^{N}\\rho_i\\ .\n",
     "\\end{eqnarray}\n",
     "$$\n",
-    "For an observable Pauli string $O=\\bigotimes_{j=1}^{n}P_j,\\ P_j\\in\\{\\mathbb{I}, X, Y, Z\\}$, we can directly use $\\rho$ to calculate $\\langle O\\rangle=\\text{Tr}(O\\rho)$. In practice, we will divide the classical shadows into $K$ parts to calculate the expectation value independently and take the median to avoid the influence of outliers:\n",
+    "For an observable Pauli string $O=\\bigotimes_{j=1}^{n}P_j,\\ P_j\\in\\{\\mathbb{I}, X, Y, Z\\}$, we can directly use $\\rho$ to calculate $\\langle O\\rangle=\\text{Tr}(O\\rho)$. In practice, we will divide the classical shadows into $K$ parts to calculate the expectation values independently and take the median to avoid the influence of outliers:\n",
     "$$\n",
     "\\begin{equation}\n",
     "    \\langle O\\rangle=\\text{median}\\{\\langle O_{(1)}\\rangle\\cdots\\langle O_{(K)}\\rangle\\},\n",
@@ -176,7 +176,7 @@
     "\n",
     "epsilon, delta = 0.1, 0.01\n",
     "N, K = shadows.shadow_bound(ps, epsilon, delta)\n",
-    "nps = N // r  # number of random pauli strings\n",
+    "nps = N // r  # number of random selected Pauli strings\n",
     "print(f\"N: {N}\\tK: {K}\\tnumber of Pauli strings: {nps}\")"
    ],
    "metadata": {
@@ -226,7 +226,7 @@
   {
    "cell_type": "markdown",
    "source": [
-    "We randomly generate Pauli strings. Since the function after just-in-time compilation does not support random sampling, we need to generate all random states in advance, that is, variable `status`."
+    "We randomly generate Pauli strings. Since the function after just-in-time (jit) compilation does not support random sampling, we need to generate all random states in advance, that is, variable `status`."
    ],
    "metadata": {
     "collapsed": false
@@ -328,7 +328,7 @@
   {
    "cell_type": "markdown",
    "source": [
-    "It can be seen from the running time that every time the number of Pauli strings changes, `shadow_expec` will be recompiled, but for the same number of Pauli strings but different observables, `shadow_expec` will only be compiled once. In the end, the absolute errors given by classical shadows are much smaller than the $\\epsilon=0.1$ we set, so shadow_bound gives a very loose upper bound."
+    "It can be seen from the running time that every time the number of Pauli strings changes, `shadow_expec` will be recompiled, but for the same number of Pauli strings but different observables, `shadow_expec` will only be compiled once. In the end, the absolute errors given by classical shadows are much smaller than the $\\epsilon=0.1$ we set, so `shadow_bound` gives a very loose upper bound."
    ],
    "metadata": {
     "collapsed": false
@@ -552,14 +552,14 @@
   {
    "cell_type": "markdown",
    "source": [
-    "On the other hand, for the second order Renyi entropy, we have another method to calculate it in polynomial time by random measurement:\n",
+    "On the other hand, for the second order Renyi entropy, we have another method to calculate it in polynomial time by random measurements:\n",
     "$$\n",
     "\\begin{eqnarray}\n",
     "    S_2&=&-\\log\\left(\\text{Tr}(\\rho_A^2)\\right),\\\\\n",
     "    \\text{Tr}(\\rho_A^2)&=&2^k\\sum_{b,b'\\in\\{0,1\\}^k}(-2)^{-H(b,b')}\\overline{P(b)P(b')},\n",
     "\\end{eqnarray}\n",
     "$$\n",
-    "where $A$ is the $k$-d reduced system, $H(b,b')$ is the Hamming distance between $b$ and $b'$, $P(b)$ is the probability for measuring $\\rho_A$ and obtaining the outcomes $b$ thus we need a larger $r$ to obtain a good enough priori probability, and the overline means the average on all random selected Pauli strings. Please refer to [Brydges, et al. (2019)](https://www.science.org/doi/full/10.1126/science.aau4963) for more details. We can use `renyi_entropy_2` to implement this method, but it is not jitable because we need to build the dictionary based on the bit strings obtained by measurement. Compared with `entropy_shadow`, it cannot filter out non-negative eigenvalues, so the accuracy is slightly worse."
+    "where $A$ is the $k$-d reduced system, $H(b,b')$ is the Hamming distance between $b$ and $b'$, $P(b)$ is the probability for measuring $\\rho_A$ and obtaining the outcomes $b$ thus we need a larger $r$ to obtain a good enough priori probability, and the overline means the average on all random selected Pauli strings. Please refer to [Brydges, et al. (2019)](https://www.science.org/doi/full/10.1126/science.aau4963) for more details. We can use `renyi_entropy_2` to implement this method, but it is not jitable because we need to build the dictionary based on the bit strings obtained by measurements, which is a dynamical process. Compared with `entropy_shadow`, it cannot filter out non-negative eigenvalues, so the accuracy is slightly worse."
    ],
    "metadata": {
     "collapsed": false
@@ -608,7 +608,7 @@
   {
    "cell_type": "markdown",
    "source": [
-    "We can `global_shadow_state`, `global_shadow_state1` or `global_shadow_state2` to reconstruct the density matrix. These three functions use different methods, but the results are exactly the same and all of them are jitable. In specific, `global_shadow_state` uses `kron` and is recommended, the other two use `einsum`."
+    "We can use `global_shadow_state`, `global_shadow_state1` or `global_shadow_state2` to reconstruct the density matrix. These three functions use different methods, but the results are exactly the same and all of them are jitable. In specific, `global_shadow_state` uses `kron` and is recommended, the other two use `einsum`."
    ],
    "metadata": {
     "collapsed": false