Update shadows workflow images (#2142)

* typos in user guide and examples

* Fix #2113 (comment)

* min's feedback - one workflow is for calibration

docs/source/examples/rshadows_tutorial.md

@@ -35,7 +35,10 @@ from mitiq.interface.mitiq_cirq.cirq_utils import (
 np.random.seed(666)
 ```
 
-Whether to run the quantum measurement or directly use the results from the previous run. If **True**, the measurement will be run again. If **False**, the results from the previous run will be used.
+There are two options: Whether to run the quantum measurement or directly use the results from the previous run. 
+
+- If **True**, the measurement will be run again.
+- If **False**, the results from the previous run will be used.
 
 
 ```{code-cell} ipython3
@@ -55,7 +58,7 @@ if not run_quantum_processing:
         saved_data = pickle.load(zf.open(f"{saved_data_name}.pkl"))
 ```
 
-The *robust shadow estimation*{cite}`chen2021robust` approach put forth in *Predicting Many Properties of a Quantum System from Very Few Measurements* {cite}`huang2020predicting` exhibits noise resilience. The inherent randomization in the protocol simplifies the noise, transforming it into a Pauli noise channel that can be characterized relatively straightforwardly. Once the noisy channel $\widehat{\mathcal{M}}$ is characterized $\widehat{\mathcal{M}}$, it is incorporated into the channel inversion $\widehat{\mathcal{M}}^{-1}$, resulting in an unbiased state estimator. The sampling error in the determination of the Pauli channel contributes to the variance of this estimator. 
+The *robust shadow estimation*{cite}`chen2021robust` approach based on {cite}`huang2020predicting` exhibits noise resilience. The inherent randomization in the protocol simplifies the noise, transforming it into a Pauli noise channel that can be characterized relatively straightforwardly. Once the noisy channel $\widehat{\mathcal{M}}$ is characterized, it is incorporated into the channel inversion $\widehat{\mathcal{M}}^{-1}$, resulting in an unbiased state estimator. The sampling error in the determination of the Pauli channel contributes to the variance of this estimator. 
 
 ## 1. Define Quantum Circuit and Executor
 In this notebook, we'll use the ground state of the Ising model with periodic boundary conditions as an example to study energy and two-point correlation function estimations. We'll compare the performance of robust shadow estimation with the standard shadow protocol, taking into account the bit-flip or depolarization noise on the quantum channels.
@@ -96,7 +99,7 @@ else:
     circuit = circuit.transform_qubits(qubit_map=qubit_map)
 ```
 
-similar with the classical shadow protocal, we define the executor to perform the computational measurement for the circuit. Here we use the simulator that adds single-qubit depolarizing noise after rotation gates and before the $Z$-basis measurement, as the noise is assumped to be gate independent, time invariant and Markovian, the noisy gate satisfies $U_{\Lambda_U}(M_z)_{\Lambda_{\mathcal{M}_Z}}\equiv U\Lambda\mathcal{M}_Z$:
+Similar with the classical shadow protocol, we define the executor to perform the computational measurement for the circuit. Here, we use add single-qubit depolarizing noise after rotation gates but before the $Z$-basis measurement. As the noise is assumed to be gate independent, time invariant and Markovian, the noisy gate satisfies $U_{\Lambda_U}(M_z)_{\Lambda_{\mathcal{M}_Z}}\equiv U\Lambda\mathcal{M}_Z$:
 
 
 ```{code-cell} ipython3
@@ -213,7 +216,7 @@ counts = {bitstring: bitstring.count("1") for bitstring in bitstrings}
 order = np.argsort(list(counts.values()))
 reordered_bitstrings = bitstrings[order]
 
-# solve for theoretical Pauli fidelitys
+# solve for theoretical Pauli fidelities
 f_theoretical = {}
 bitstrings = list(f_est_results.keys())
 for bitstring in bitstrings:
@@ -228,14 +231,14 @@ import matplotlib.pyplot as plt
 import seaborn as sns
 
 sns.set_style("whitegrid")
-# plot estimated vs theoretical Pauli fidelitys when no errors in quantum circuit
+# plot estimated vs theoretical Pauli fidelities when no errors in quantum circuit
 plt.plot(
     [np.abs(f_est_results[b]) for b in reordered_bitstrings],
     "-*",
     label="Noisy Channel",
 )
 plt.plot(
-    [f_theoretical[b] for b in reordered_bitstrings], label="Noisless Channel"
+    [f_theoretical[b] for b in reordered_bitstrings], label="Noiseless Channel"
 )
 plt.xlabel(r"measurement basis states $b$")
 plt.xticks(
@@ -256,7 +259,7 @@ The expectation value for a series of operators, denoted as $\{O_\iota\}_{\iota\
 \hat{o}_\iota &= \langle\!\langle O_\iota|{\hat{\rho}}\rangle\!\rangle \simeq \langle\!\langle O_\iota|\widehat{\mathcal{M}}^{-1}\widetilde{\mathcal{M}}|\rho\rangle\!\rangle=\sum_{b^{(1)}\in\{0,1\}^{n}}f_{b^{(1)}}^{-1}\left(\bigotimes_{i=1}^n \langle\!\langle P_i|\Pi_{b_i^{(1)}}\widehat{\mathcal{M}}_{P_i}\right)|\rho\rangle\!\rangle\nonumber\\ 
 &=\sum_{b^{(1)}\in\{0,1\}^{n}}f_{b^{(1)}}^{-1}\prod_{i=1}^n \langle\!\langle P_i|\Pi_{b^{(1)}_i}\bigg|U_i^{(2)\dagger}|b_i^{(2)}\rangle\langle b_i^{(2)}|U_i^{(2)}\bigg\rangle\!\bigg\rangle
 \end{align}
-where in the last equality, $\{P_i\}_{i\in n}$ represents Pauli operators, with $P=\{I,X,Y,Z\}$. And as we did previously, we use the lable $(1)$ as the subscript to distinguish the parameters of the calibration process from the parameters of the shadow estimation process, which is labelled by $(2)$. It is assumed that $O_\iota$ are Pauli strings acting on $supp(O_\iota)$ ($|supp(O_\iota)|\leq n$) sites of the system. It can be verified that the cross product over qubit sites within the summation of the final expression in the above equation is zero, except when all sites in $supp(O_\iota)^c$ have $\Pi_0$ acting on. Similarly, it can be verified that the cross product over qubit sites within the summation of the final expression in the above equation is zero, except when all sites in $supp(O_\iota)$ have $\Pi_1$ acting on, i.e.
+where in the last equality, $\{P_i\}_{i\in n}$ represents Pauli operators, with $P=\{I,X,Y,Z\}$. And as we did previously, we use the label $(1)$ as the subscript to distinguish the parameters of the calibration process from the parameters of the shadow estimation process, which is labelled by $(2)$. It is assumed that $O_\iota$ are Pauli strings acting on $supp(O_\iota)$ ($|supp(O_\iota)|\leq n$) sites of the system. It can be verified that the cross product over qubit sites within the summation of the final expression in the above equation is zero, except when all sites in $supp(O_\iota)^c$ have $\Pi_0$ acting on. Similarly, it can be verified that the cross product over qubit sites within the summation of the final expression in the above equation is zero, except when all sites in $supp(O_\iota)$ have $\Pi_1$ acting on, i.e.
 \begin{align}
 \Pi_i|I\rangle\!\rangle\equiv\delta_{i,0}|I\rangle\!\rangle,\qquad \Pi_{i}|P\rangle\!\rangle\equiv\delta_{i,1}|P\rangle\!\rangle,\qquad ~~~ \mathrm{for}~i\in\{0,1\};~P\;=\{X,\;Y,\;Z\}.
 \end{align}
@@ -265,13 +268,13 @@ Therefore, the final expression of the expectation value estimation can be simpl
 \hat{o}_\iota = \left(\sum_{b^{(1)}\in\{0,1\}^{n}}f_{b^{(1)}}^{-1}\prod_{j\in supp(O_\iota)}
 \delta_{b_j^{(1)},1}\prod_{k\in supp(O_\iota)^c}\delta_{b_k^{(1)},0}\right)\prod_{i=1}^n  \langle b_i^{(2)}|U_i^{(2)}P_i U_i^{(2)\dagger}|b_i^{(2)}\rangle
 \end{align}
- Additionally, when $P_i =X_i$ (r.e.s.p. $Y_i,\;Z_i$), $U_i^{(2)}$ must correspond to $X$ (r.e.s.p. $Y,\;Z$)-basis measurement to yield a non-zero value, which is easy to check considered that the $P$-basis measurement channel has a PTM rep: $\widetilde{\mathcal{M}}_{P}=\frac{1}{2}(|I\rangle\!\rangle\langle\!\langle I|+|P\rangle\!\rangle\langle\!\langle P|)$, observiously the only measurement that didn't vanished by the operator's $i$-th qubit component in PTM rep: $P\rightarrow \langle\!\langle P|$, is the local $P$-basis measurement.  
+ Additionally, when $P_i =X_i$ (r.e.s.p. $Y_i,\;Z_i$), $U_i^{(2)}$ must correspond to $X$ (r.e.s.p. $Y,\;Z$)-basis measurement to yield a non-zero value, which is easy to check considered that the $P$-basis measurement channel has a PTM rep: $\widetilde{\mathcal{M}}_{P}=\frac{1}{2}(|I\rangle\!\rangle\langle\!\langle I|+|P\rangle\!\rangle\langle\!\langle P|)$. Obviously, the only measurement that didn't vanish by the operator's $i$-th qubit component in PTM rep: $P\rightarrow \langle\!\langle P|$, is the local $P$-basis measurement.  
 
-Next steps are same with classical protocol,with the statistical method of taking an average called "median of means" to achieve an acceptable failure probability of estimation, which need $R_2=N_2K_2$ snapshots, where we use the subscript "2" to denote the index of classical shadow protocol. Acctually,
+Next steps are identical to the classical protocol, where the statistical method of taking an average called "median of means" is used to achieve an acceptable failure probability of estimation. This needs $R_2=N_2K_2$ snapshots, where we use the subscript "2" to denote the index of classical shadow protocol. Actually,
 \begin{align}
 \hat{o}_\iota(N_2,K_2):=\mathrm{median}\{\hat{o}_\iota^{(1)},\cdots,\hat{o}_\iota^{(K_2)}\}~~\mathrm{where}~~\hat{o}_\iota^{(j)}=N_2^{-1}\sum_{k=N_2(j-1)+1}^{N_2j}\mathrm{Tr}(O_\iota\hat{\rho}_k),\qquad \forall~1\leq j\leq K_2,
 \end{align}
-where we have $K_2$ estimators each of which is the average of $N_2$ single-round estimators $\hat{o}_i^{(j)}$, and take the median of these $K_2$ estimators as our final estimator $\hat{o}_\iota(N_2,K_2)$. We can calculate the median of means of each irreps with projection $\Pi_b=\bigotimes_{i=1}^n\Pi_{b_i}$, 
+where we have $K_2$ estimators each of which is the average of $N_2$ single-round estimators $\hat{o}_i^{(j)}$, and take the median of these $K_2$ estimators as our final estimator $\hat{o}_\iota(N_2,K_2)$. We can calculate the median of means of each irreducible representations with projection $\Pi_b=\bigotimes_{i=1}^n\Pi_{b_i}$, 
 
 ### 3.1 Ground State Energy Estimation of Ising model with the RSHADOWS algorithm
 

docs/source/examples/shadows_tutorial.md

@@ -15,7 +15,8 @@ kernelspec:
 
 **Corresponding to:** Min Li (minl2@illinois.edu)
 
-This notebook shows how to use classical shadows estimation with the Mitiq library, focused initially on local (Pauli) measurements. We show some common scenarios such as state tomography, and operator expectation value estimation. The method creates an approximate classical description of a quantum state with few measurements while effectively characterizing and mitigating noise in the following notebook `rshadows_tutorial`.
+This notebook shows how to use classical shadows estimation with the Mitiq library, focused initially on local (Pauli) measurements. We show some common scenarios such as state tomography, and operator expectation value estimation. The method creates an approximate classical description of a quantum state with few measurements while effectively characterizing and mitigating noise in the [following notebook](https://mitiq.readthedocs.io/en/stable/examples/rshadows_tutorial.html).
+
 ```{code-cell} ipython3
 import cirq
 import numpy as np
@@ -75,12 +76,12 @@ MathJax.Hub.Config({
 
 ## 2. Sampling random Pauli measurements 
 
-This process involves applying a random unitary selected from a randomly fixed ensemble $\mathcal{U}\in U(2^n)$ to rotate the state $\rho\rightarrow U^\dagger \rho U$, performing a computational-basis($Z$-basis) measurement, and storing a classical description $U^\dagger |\hat{b}\rangle\langle\hat{b}| U$. After the measurement, the inverse of $U$ is applied to the resulting computational basis state, collapsing $\rho$ to
+This process involves applying a random unitary selected from a randomly fixed ensemble $\mathcal{U}\in U(2^n)$ to rotate the state $\rho\rightarrow U^\dagger \rho U$, followed by a computational-basis($Z$-basis) measurement, and storing a classical description $U^\dagger |\hat{b}\rangle\langle\hat{b}| U$. After the measurement, the inverse of $U$ is applied to the resulting computational basis state, collapsing $\rho$ to
 
 \begin{equation}
 U^\dagger|\hat{b}\rangle\langle\hat{b}| U\qquad \mathrm{where} \qquad \mathrm{Pr}[\hat{b}=b]=\langle b|U\rho U^\dagger|b\rangle.
 \end{equation}
-If the unitary group $\mathcal{U}$ is chosen to be the local Clifford group $\mathrm{CL}(2)^n$, this equavelent to performing a random Pauli measurement on each qubit. This means that for each qubit, we randomly decide to measure one of the Pauli operators. Define the `cirq_executor` take one shot of measurement and return the measurement result.
+If the unitary group $\mathcal{U}$ is chosen to be the local Clifford group $\mathrm{CL}(2)^n$, this equivalent to performing a random Pauli measurement on each qubit. This means that for each qubit, we randomly decide to measure one of the Pauli operators. Below, we define the `cirq_executor` to take one shot of measurement and return the measurement result.
 
 
 ```{code-cell} ipython3
@@ -95,9 +96,9 @@ def cirq_executor(
     )
 ```
 
-In terms of implementation, considering that the only possible measurement to be performed is the $Z$-basis measurement, the random Pauli measurement is equivalent to randomly sampling a unitary from the unitary ensemble: $\mathcal{G}=\{\mathrm{id},\mathrm{H},\mathrm{H}\cdot \mathrm{S}^\dagger\}$. Afterward, the $Z$-basis measurement is conducted. We then record a sequence of Pauli gates $u_i:= U_i^\dagger ZU_i$ that have been measured for each qubit in the circuit. This sequence becomes one of the output lists of the measurement function random_pauli_measurement.
+In terms of implementation, considering that the only possible measurement to be performed is the $Z$-basis measurement, the random Pauli measurement is equivalent to randomly sampling a unitary from the unitary ensemble: $\mathcal{G}=\{\mathrm{id},\mathrm{H},\mathrm{H}\cdot \mathrm{S}^\dagger\}$. Afterward, the $Z$-basis measurement is conducted. We then record a sequence of Pauli gates $u_i:= U_i^\dagger ZU_i$ that have been measured for each qubit in the circuit. This sequence becomes one of the output lists of the measurement function `random_pauli_measurement`.
 
-In the main function, the quantum measurement process is encapsulated within the shadow_quantum_processing function. This function takes the quantum circuit and the number of shots as input. It returns the measurement results as bit strings, for example, '01...0' is equivelent to the measurement basis eigenstate: $|0\rangle|1\rangle...|0\rangle$. Additionally, it provides the measured Pauli gates in string format. For instance, 'XY...Z' signifies a local X-basis measurement on the first qubit, a local Y-basis measurement on the second qubit, and a local Z-basis measurement on the last qubit in the circuit.
+In the main function, the quantum measurement process is encapsulated within the `shadow_quantum_processing function`. This function takes the quantum circuit and the number of shots as input. It returns the measurement results as bit strings, for example, '01...0' is equivalent to the measurement basis eigenstate: $|0\rangle|1\rangle...|0\rangle$. Additionally, it provides the measured Pauli gates in string format. For instance, 'XY...Z' signifies a local X-basis measurement on the first qubit, a local Y-basis measurement on the second qubit, and a local Z-basis measurement on the last qubit in the circuit.
 
 
 ```{code-cell} ipython3
@@ -123,7 +124,7 @@ which has been named a single copy of **classical shadow**. Based on *Schur's Le
 \end{equation}
  which is indeed unitary, however, not CP, so it is not a physical map as expected.
 
-In the case of random Pauli measurement, the unitary could be represented by the tensor product of all qubits, so it is with the state $|\hat{b}\rangle\in\{0,1\}^{\otimes n}$, i.e. $U^\dagger|\hat{b}\rangle=\bigotimes_{i\leq n}U_i|\hat{b}_i\rangle$. Therefore, based on Schur's Lemma, a snapshot would takes the form:
+In the case of random Pauli measurement, the unitary could be represented by the tensor product of all qubits, so it is with the state $|\hat{b}\rangle\in\{0,1\}^{\otimes n}$, i.e. $U^\dagger|\hat{b}\rangle=\bigotimes_{i\leq n}U_i|\hat{b}_i\rangle$. Therefore, based on Schur's Lemma, a snapshot would take the form:
 \begin{equation}
 \hat{\rho}=\bigotimes_{i=1}^{n}\left(3U_i^\dagger|\hat{b}_i\rangle\langle\hat{b}_i|U_i-\mathbb{I}\right),\qquad|\hat{b}_i\rangle\in\{0,1\}.
 \end{equation}
@@ -141,24 +142,24 @@ The classical shadows state reconstruction are then obtained by taking the avera
 this is realized in the function `state_reconstruction`. In the main function `classical_post_processing`, we take the output of `shadow_quantum_processing`, then apply the inverse channel to obtain the snapshots, and finally take the average of the snapshots to obtain the reconstructed state if *state_reconstruction =* **True**. **In the current notebook, we don't preform Pauli twirling calibration, and we set** *rshadow* = **False**.
 
 #### 4.1.1 Error Analysis
-We can take a visualization of the elementwise difference between the reconstructed state and the original state. 
+We can take a visualization of the element wise difference between the reconstructed state and the original state. 
 \begin{equation}
 \Delta\rho_{ij}=|\rho^{\mathrm{shadow}}_{ij}-\rho_{ij}|
 \end{equation}
-The difference is very small, which means that the classical shadow is a good approximation of the original state even in the sence of state tomography. 
+The difference is very small, which means that the classical shadow is a good approximation of the original state even in the sense of state tomography. 
 
 It is anticipated that the fidelity will not necessarily be lower than 1, as the state reconstructed through classical shadow estimation is not guaranteed to be a physical quantum state, given that $\mathcal{M}^{-1}$ is not a quantum channel. 
 
 Fidelity is defined by $F(\rho,\sigma)=\mathrm{Tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$, when $\rho=|v\rangle\langle v|$ is a pure state $F(\rho,\sigma)=\langle v|\sigma|v\rangle$.
-Based on the theorem, if the error rate of fedelity is $\epsilon$, i.e.
+Based on the theorem, if the error rate of fidelity is $\epsilon$, i.e.
 \begin{equation}
 |F(\rho,\sigma)-1|\leq\epsilon,
 \end{equation}
 then the minimum number of measurements $N$ (number of snapshots) should be:
 \begin{equation}
 N = \frac{34}{\epsilon^2}\left\|\rho-\mathrm{Tr}(\rho)/{2^n}\mathbb{I}\right\|_{\mathrm{shadow}}^2
 \end{equation}
-with the shadow norm upper bound of the random Pauli measurement $\left\|\cdot\right\|_{\mathrm{shadow}}\leq 2^k\|\cdot\|_\infty$ when the operator acting on $k$ qubits, we have $N\leq 34\epsilon^{-2}2^{2n}+\mathcal{O}(e^{-n})$. Based on Fuchs–van de Graaf inequalities and properties of $L_p$ norm, $\|\rho-\sigma\|_2\leq \|\rho-\sigma\|_1 \leq (1-F(\rho,\sigma))^{1/2}$, the $L_2$ norm distance between the state reconstructed through classical shadow estimation and the state prepared by the circuit is upperbounded by the fidelity error rate $\epsilon$. The dependency of the bound number of measurements $N$ to achieve the error rate $\epsilon$ is depicked in function `n_measurements_tomography_bound`.
+with the shadow norm upper bound of the random Pauli measurement $\left\|\cdot\right\|_{\mathrm{shadow}}\leq 2^k\|\cdot\|_\infty$ when the operator acting on $k$ qubits, we have $N\leq 34\epsilon^{-2}2^{2n}+\mathcal{O}(e^{-n})$. Based on Fuchs–van de Graaf inequalities and properties of $L_p$ norm, $\|\rho-\sigma\|_2\leq \|\rho-\sigma\|_1 \leq (1-F(\rho,\sigma))^{1/2}$, the $L_2$ norm distance between the state reconstructed through classical shadow estimation and the state prepared by the circuit is upperbound by the fidelity error rate $\epsilon$. The dependency of the bound number of measurements $N$ to achieve the error rate $\epsilon$ is depicted in function `n_measurements_tomography_bound`.
 
 
 ```{code-cell} ipython3
@@ -186,13 +187,13 @@ rho_shadow = output["reconstructed_state"]
 
 
 ```{code-cell} ipython3
-# Compute the ideal state vector discribed by the input circuit.
+# Compute the ideal state vector described by the input circuit.
 state_vector = test_circuits.final_state_vector().reshape(-1, 1)
 # Compute the density matrix.
 rho_true = state_vector @ state_vector.conj().T
 ```
 
-We can plot the elementwise difference between the reconstructed state and the original state as a thermal diagram:
+We can plot the element wise difference between the reconstructed state and the original state as a thermal diagram:
 
 
 ```{code-cell} ipython3