Neural-shadow quantum state tomography (2024)

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  Figure 1

  Overview of the NNQST and NSQST protocols. Panel (a)shows the NNQST protocol with the cross-entropy loss function $L_{\lambda }$. The training data determine $p_{\mathrm{\Phi }}(s,B)$, the measured probability distribution of measurement outcomes $s$ for measurements of the target state $|\mathrm{\Phi }\rangle$ performed in the local Pauli basis $B$. The feedback loop on the right-hand side indicates the iterative first-order optimization for neural network training. Panel (b)displays the NSQST protocol described in Sec.2d, where the training data set consists of classical shadows only and where the network parameters $\lambda$ are trained via an infidelity loss function ${\mathcal{L}}_{\lambda }$. The expression ${\widehat{\rho }}_i(U_i^{†},b_i)$ is the stored classical shadow of the target state $|\mathrm{\Phi }\rangle$ with the Clifford unitary $U_i^{†}$ and bit-string $|b_i\rangle$.

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  Figure 2

  Overview of the NSQST with pretraining protocol. The neural networks learning $p_{{\lambda }_1}\left(s\right)$ and ${\varphi }_{{\lambda }_2}\left(s\right)$ are separately parameterized, and $p_{{\lambda }_1}\left(s\right)$ is pretrained using NNQST with training data derived from measurements only in the computational basis.

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  Figure 3

  Tomography of the quantum state following a one-dimensional QCD time evolution. Two Trotter steps are used for a total evolution time of $t=1.8$. Panel (a)displays the average final-state infidelity for each of the three protocols. In each trial, we extract the (exactly calculated) average state infidelity ${\mathcal{L}}_{\lambda }$, averaged over the last 100 iterations (and further averaged over ten trials). The error bar is the standard error in the mean calculated over the ten trials. The embedded schematic shows the qubit encoding for a unit cell, containing up to three quarks (filled circles) and up to three antiquarks (striped circles). In panel (b), the plot compares the expectation value of the kinetic energy, evaluated for the neural network quantum state found in the last iteration of each trial, and averaged over ten trials for each protocol. In panel c, the optimization progress curves are displayed for a typical trial, where the adjusted iteration refers to epochs for NNQST, but rather indicates increments of ten iterations for the two NSQST protocols (a total of 2000 iterations were run in these cases). Panel (c)shows the NNQST (blue) loss $L_{\lambda }$ in the top plot, the estimated NSQST (infidelity) loss function ${\mathcal{L}}_{\lambda }$ (with and without pretraining, green and red, respectively) in the middle plot (fluctuations are dominated by the finite number $N=100$ of classical shadows taken for each estimate), and the exact infidelity is shown in every (adjusted) iteration for all three protocols in the lower plot.

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  Figure 4

  Typical neural network quantum states following optimization, approximating the state after a one-dimensional QCD time evolution. Panel (a)displays a typical state found in the last iteration of NNQST training: the left plot shows the square root of the probability of the final neural network quantum state compared to the exact target state and the right plot shows the phase output of the state over the set of computational basis states $s$. To highlight the dominant contributions, the phase output has been truncated for states $s$ with $\sqrt{p_{{\lambda }_1}\left(s\right)}<0.1$. For the purpose of better visualization, the overall (global) phase of the neural network quantum state is chosen by aligning the phase of the most probable computational-basis state to that of the target state. The dashed line corresponds to a phase of $2\pi$, since we choose our phase predictions to be in the range $[0,2\pi ]$. Panels (b)and (c)show typical final states from NSQST and NSQST with pretraining. We observe that both NSQST protocols succeed at learning the phase structure while NNQST fails at the same task.

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  Figure 5

  Training and results for a simulation of tomography on the time-evolved state for a one-dimensional AFH model. Four Trotter steps are used for a total evolution time of $t=0.8$. Panel (a)compares the final state infidelity, averaged over ten trials for the three protocols, following the same procedure used for one-dimensional QCD time evolution. Panel (b)compares the predicted mean staggered magnetization in the $x$ direction (where $S_j^x=\frac{1}{2}{\sigma }_j^x$), following a Trotterized time evolution under the AFH model, for all the three protocols. In panel (c), we show optimization progress curves for a typical run, with the NNQST (blue) loss $L_{\lambda }$ in the top plot, NSQST loss functions (with and without pretraining, green and red, respectively) in the middle plot, and the exact infidelity in every adjusted iteration for all three protocols in the lower plot.

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  Figure 6

  Typical final neural network quantum states, trained on the time-evolved state of a one-dimensional AFH model. Panel (a)displays a typical final state in the last iteration of NNQST optimization, generated using the same procedure from Fig.4. Panels (b)and (c)show the typical final states from NSQST and from NSQST with pretraining, respectively.

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  Figure 7

  Simulation of tomography on a six-qubit phase-shifted GHZ state. Panel (a)compares the final state infidelity, averaged over ten trials for each of the three protocols. Panel (b)shows typical optimization progress curves for NNQST (blue), NSQST (red), and NSQST with pretraining (green).

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  Figure 8

  Comparison of NSQST with pretraining to direct shadow estimation. In each trial of NSQST with pretraining, 200 Clifford shadows are used as training data without resampling in every iteration and 1000 measurements in the computational basis are used in pretraining. For direct shadow estimation, 1200 Clifford shadows and 1200 Pauli shadows are used. Panel (a)compares the absolute error in the predicted kinetic energy, averaged over ten trials for each of the four protocols. Panel (b)compares the absolute error in the predicted fidelity to the ideal time-evolved state, averaged over ten trials for each of the four protocols. Panel (c)compares the absolute error in the predicted expectation value of a Pauli string observable of various weight in the Pauli-X basis, averaged over ten trials for each of the four protocols. The data points are slightly shifted relative to the ticks of the $x$ axis for a better display of error bars.

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  Figure 9

  Infidelity and predicted expectation value of the phase-shifted GHZ state as the system size grows. For each system size, we generate 3000 computational-basis measurements and 200 Clifford shadows. In panel (a), with independently sampled measurement data and 5000 Monte Carlo samples, we run NSQST with pretraining using the improved strategy for five trials and report the individual final infidelities. Note that the number of Monte Carlo samples used during training is much less than the number of basis states, which is not an issue if the target state is sufficiently sparse, as in the case of the multiqubit GHZ state. In panel (b), we plot the predicted expectation value of the Pauli string $XX...XY$, which is one of the target state's stabilizers. In comparison, the expectation values predicted from five trials of direct shadow estimation are plotted, each with 3200 independently sampled Clifford or Pauli shadows. The data points are slightly shifted relative to the ticks of the $x$ axis for a better display.

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  Figure 10

  Simulation of tomography for a phase-shifted GHZ state in the presence of noise. Panel (a)displays the average loss function (red) defined in Eq.(23) and the exact infidelity (blue) for the amplitude damping channel ${\mathrm{AD}}_{n,p}$ (the loss function is averaged over the last 100 iterations for each trial and then the average is taken over ten trials). The strength of the noise increases with increasing $1-p$. The noiseless infidelity loss function $\mathcal{L}\left(\mathbb{I}\right)$ is then transformed into an estimated infidelity for the noisy case $\mathcal{L}\left(\mathcal{E}\right)$ using Eq.(22) for the amplitude damping channel $\mathcal{E}={\mathrm{AD}}_{n,p}$ to obtain the transformed cost function. The error bars represent the standard error in the mean over ten trials. In panel (b), we show the average loss function and exact infidelity for the local depolarizing noise model with a two-qubit depolarizing channel applied after every CNOT gate in the appended random Clifford circuit $U_i$. The channel parameter $1-f$ characterizes the growing strength of the noise. We do not plot the transformed loss function in this case because the CNOT-dependent local depolarizing noise model does not have an analytic noisy shadow expression.

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  Figure 11

  Simulation circuit for the one-dimensional QCD model. The initial state preparation circuit (before the barrier) and a single Trotter step (after the barrier) are drawn using Qiskit [69]. In our numerical experiments an evolution for time $t=1.8$ is decomposed into two Trotter steps.

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  Figure 12

  Simulation circuit for the one-dimensional AFH model. The initial state preparation circuit (before the barrier) and a single Trotter step (after the barrier) are drawn using Qiskit [69]. In our numerical experiments an evolution for time $t=0.8$ is decomposed into four Trotter steps.

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  Figure 13

  Additional plots for the quantum state following a one-dimensional QCD time evolution. Panel (a)displays the expectation values of the total energy, evaluated for the neural network quantum state found in the last iteration of each trial, and averaged over ten trials for each protocol. Panel (b)displays the expectation values of the mass Hamiltonian.

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  Figure 14

  Typical optimization progress curve from NSQST with pretraining and fixed Clifford shadows. Unlike the other plots, the iteration number on the $x$ axis is not adjusted and corresponds to every gradient update during optimization.

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