Neural–Shadow Quantum State Tomography

Our pure-state neural network quantum state ansatz is adopted from Ref. [18]. The parameterized model is based on the transformer architecture [15], widely used in natural language processing and computer vision [25, 26]. As compared to older architectures such as the RBM, the transformer is superior in modelling long-range interactions and allows for more efficient sampling of the encoded probability distribution due to its autoregressive property [18].

The transformer neural network quantum state ansatz takes a bit-string $s=(s_{1},\ldots,s_{n})\in\{0,1\}^{n}$ corresponding to the computational basis state $|s\rangle$, and produces a complex-valued amplitude $\braket{s|\psi_{\lambda}}=\psi_{\lambda}(s)$ parameterized by ${\lambda}=({\lambda_{1}},{\lambda_{2}})$ as

where $\lambda_{1}$ and $\lambda_{2}$ are vectors of real-valued model parameters for the normalized probability amplitudes $p_{\lambda_{1}}(s)$ and the phases $\varphi_{\lambda_{2}}(s)$ of the neural network quantum state. These amplitudes and phases may be parameterized by the neural network quantum state in various fashions. One approach is to use two completely disjoint models, independently parameterized by $\lambda_{1}$ and $\lambda_{2}$ for the amplitude and phase values, respectively [9, 27]. Another approach is to use a single model parameterized by $\lambda$ to encode both the amplitude and phase outputs, either via complex-valued model parameters [17, 28] or by using real-valued model parameters with two disjoint layers of output neurons connected to a common preceding neural network [18]. In our numerical experiments, we use the later parameterization for NNQST and NSQST, but the modified NSQST protocol with pre-training in Sec.˜II.5 uses two separately parameterized neural networks. See Appendix˜A for a more detailed account of the transformer architecture.

Given a trained neural network quantum state, observables and other state properties of interest can be predicted by drawing (classical) samples from the neural network model. The number of samples required to predict the expectation value of an arbitrary Pauli string (independent of its weight) and fidelity to a computationally tractable state with bounded additive error is independent of the system size [29]. Computationally tractable states include stabilizer states and neural network quantum states. Thus, if sampling can be performed efficiently, the prediction errors from neural network quantum states are primarily due to imperfect training. We also note that not every neural network quantum state ansatz has this property of efficient observable and fidelity prediction, where an important example is a class of generative models trained on informationally complete positive-operator valued measures (IC-POVMs) [30, 31].

NNQST (Fig.˜1a) aims at obtaining a trained neural network representation that closely approximates an unknown target quantum state. The training is done by iteratively adjusting the neural network parameters along a loss gradient estimated from the measurement samples in various local Pauli bases (obtained by applying single-qubit rotations before performing measurements in the computational basis) [18]. We denote a local Pauli basis as $B=(P_{1},P_{2},\cdots,P_{n})$, with $P_{i}\in\{X,Y,Z\}$. If a measurement sample $s\in\{0,1\}^{n}$ is obtained after performing rotations to the Pauli basis $B$, we store the pair $(s,B)$ as a training sample, corresponding to a product state $\ket{s,B}$.

After choosing a subset $\mathcal{B}$ of Pauli bases for collecting measurement samples, we estimate a loss function that represents the distance between the target state $\ket{\Phi}$ and the neural network quantum state $\ket{\psi_{\lambda}}$. The loss function in NNQST is based on the cross-entropy of the measurement outcome distributions for the target and neural network states in each basis $B$, which is then averaged over the set of bases $\mathcal{B}$. Ignoring a $\lambda$-independent contribution arising from the average entropy of the target-state measurement distribution, this procedure gives the cross-entropy loss function for NNQST [18]:

Here, $p_{\Phi}(s,B)$ is the probability of measuring the outcome $s$ from the target state $\ket{\Phi}$ after rotating to the Pauli basis $B$ and $p_{\psi_{\lambda}}(s,B)$ is defined as

where the overlap between the Pauli product state and the neural network quantum state requires a summation over the computational basis states $\ket{t}$ that satisfy $\braket{s,B|t}\neq 0$. Note that the number of these states $\ket{t}$ is $2^{K}$, with $K$ being the number of positions $i$ where $P_{i}\neq Z$. This suggests that an efficient and exact calculation of $p_{\psi_{\lambda}}(s,B)$ requires the projective measurements to be in almost-diagonal Pauli bases for a generic neural network quantum state $\ket{\psi_{\lambda}}$ [18].

Using the law of large numbers, the cross-entropy loss can be approximated via a finite training data set $\mathcal{D}_{T}$ as

An approximation for the gradient $\nabla_{\lambda}L$ is then directly found from Eq.˜4. During training, the gradient is provided to an optimization algorithm such as stochastic gradient descent (SGD) or one of its variants (e.g., the Adam optimizer [16]). In this paper, we exclusively use the Adam optimizer.

Shadow tomography relies on the ingenious observation that a polynomial number of measurement samples is sufficient to predict certain observables for quantum states of arbitrary size [22]. The classical shadow protocol further exploits the efficiency of the stabilizer formalism, making this procedure ready for practical experiments [8, 32, 33, 34]. In this paper, we focus on estimating linear observables of the form $\operatorname{Tr}(O\rho)$ for a pure state $\rho=|\Phi\rangle\!\langle\Phi|$. An important example (for $O=|\psi_{\lambda}\rangle\!\langle\psi_{\lambda}|$) is the fidelity between the target state $\ket{\Phi}$ and a reference state $\ket{\psi_{\lambda}}$. The first step in the protocol is to collect the so-called classical shadows of $\ket{\Phi}$. To obtain a single classical shadow sample, we apply a randomly-sampled Clifford unitary $U_{i}\in{\sf Cl}(2^{n})$ to the quantum state and measure all $n$ qubits in the computational basis, resulting in a single bit-string $\ket{b_{i}}$. The stabilizer states $\ket{\phi_{i}}=U_{i}^{\dagger}\ket{b_{i}}$ contain valuable information about $\rho$. Using representation theory [35], it can be shown that the density matrix obtained from an average over both the random unitaries and the measured bit-strings $\mathcal{M}(|\Phi\rangle\!\langle\Phi|):=\mathbb{E}_{U\sim{\sf Cl}(2^{n}),b\sim P_{\Phi}(b)}[|\phi\rangle\!\langle\phi|]$ coincides with the outcome of a depolarizing noise channel:

where $\mathcal{D}_{n,f}(\rho)=f\rho+(1-f)\frac{\mathbb{I}}{2^{n}}$ denotes the depolarizing noise channel of strength $f$. The original state can then be recovered as an average over classical shadows by inverting the above formula, $|\Phi\rangle\!\langle\Phi|=\mathbb{E}\left[\mathcal{M}^{-1}(|\phi\rangle\!\langle\phi|)\right]$. We emphasize that the original state can only be recovered after sampling from a prohibitively large number of Clifford unitaries, and for each of them sampling an exponentially large number of bit-strings. The classical shadows are therefore defined by [8]:

More generally, in the presence of a gate-independent, time-stationary, and Markovian noise channel $\mathcal{E}$ afflicting the segment of the circuit between the preparation of the state $\rho$ and the measurements, this definition extends to [23]:

Here, $f(\mathcal{E})$ is the strength of a depolarizing noise channel $\mathcal{D}_{n,f(\mathcal{E})}$ comprised of the combined effects of the channel in Eq.˜5 and the twirling of the additional noise by the random Clifford unitaries, effectively imposing further depolarization. Koh and Grewal [23] derived an analytic expression for $f(\mathcal{E})$ as

where

is the sum of fidelities for the noise channel $\mathcal{E}$ acting on each of the computational basis states $\ket{s}$ ($\mathrm{fid}(\mathcal{E})\in[0,2^{n}]$; at the lower bound, $\mathrm{fid}(\mathcal{E})=0$, the depolarizing parameter becomes negative, $f(\mathcal{E})<0$, but the associated depolarizing channel remains physical [23]). When the noise channel is not exactly known, extra calibration procedures are required in noise-robust classical shadow protocols [24].

Once the classical shadow samples $\{\hat{\rho}_{i}\}_{i=1}^{N}$ are collected, we calculate $\hat{o}^{(i)}\mathrel{\mathop{:}}\nobreak\mkern-1.2mu=\operatorname{Tr}(\hat{\rho}_{i}O)$ for each of the $N$ classical shadows and obtain an estimator for the observable from an average over the $N$ samples (alternatively, the median-of-means can be used to improve the success rate of the protocol; see Ref. [8] for more details). The key advantage of classical shadows is the bounded variance of observable estimations which, in turn, provides a bound on the number of classical shadow samples required to predict linear observables of the quantum state within a target precision. Indeed, as shown in Refs. [24, 23], the number of classical shadow samples $N$ required to estimate $M$ arbitrary linear observables $\{O_{j}\}_{j=1}^{M}$, up to an additive error $\varepsilon$, scales as

In the case of noise-free Clifford tails, $\mathcal{E}=\mathbb{I}$, we have $\mathrm{fid}(\mathbb{I})=2^{n}$, $f(\mathbb{I})=(1+2^{n})^{-1}$, and we recover the sample complexity $\mathcal{O}\left(\max_{1\leq j\leq M}\operatorname{Tr}(O_{j}^{2})\log M/\varepsilon^{2}\right)$ presented in [8], which is indeed independent of the system size $n$. The variance of observable estimators is also bounded as $\mathrm{Var}(\hat{o})\leq 3\operatorname{Tr}(O^{2})$ in this case and is independent of the system size.

Given a pure target state $\ket{\Phi}$, our goal in NSQST (Fig.˜1b) is to progressively adjust the model parameters ${\lambda}$, such that the associated pure state $\ket{\psi_{\lambda}}$ (see Eq.˜1) approaches $\ket{\Phi}$ during optimization. We approximate the fidelity between the model and target states using the classical shadow formalism described in Sec.˜II.3, taking $O_{\lambda}=\lvert\psi_{\lambda}\rangle\!\langle\psi_{\lambda}\rvert$ as a linear observable, averaged with respect to $\rho=\lvert\Phi\rangle\!\langle\Phi\rvert$. The number $M$ of observables we predict during optimization therefore coincides with the number of descent steps taken by the optimizer, as updating $\lambda$ changes the observable $\lvert\psi_{\lambda}\rangle\!\langle\psi_{\lambda}\rvert$ in every iteration. By collecting $N$ classical shadows, we can approximate our loss function (the infidelity) via

In the noise-free case $\mathcal{E}=\mathbb{I}$, this expression simplifies to

We see that, independent of the specific form of $\mathcal{E}$, training the model is simply equivalent to increasing the average overlap between the random stabilizer states and the model quantum state.

The next step in NSQST requires classical post-processing to estimate the overlaps $\braket{\phi_{i}|\psi_{\lambda}}$. For certain states $\ket{\psi_{\lambda}}$ (e.g., stabilizer states), the overlap can be calculated efficiently. Many states of interest do not fall into this class, leading to a potential exponential overhead. However, we can obtain a Monte-Carlo estimate of the overlap by sampling from the model quantum state $\ket{\psi_{\lambda}}$. In the model, we associate a probability

to each computational basis state $\ket{s}$. Therefore,

It is now straightforward to provide a Monte Carlo estimate of the above quantity [36] (see Appendix˜D for an alternative approach). For each sample $s$ from the neural network, we have direct access to the exact complex-valued amplitudes $\psi_{\lambda}(s)$ of the neural network quantum state in the computational basis. Moreover, we can compute the stabilizer state projections $\phi_{i}(s)=\bra{s}U_{i}^{\dagger}\ket{b_{i}}$ in $O(n^{2})$ time, in view of the Gottesman-Knill theorem [37, 38, 39]. Note that decomposing a randomly sampled Clifford operator into primitive unitary gates (e.g., with Hadamard, S, and CNOT gates) still takes $O(n^{3})$ time [39, 40]. However, this is a one-time procedure to be run for each $U_{i}$ and can be done in advance of state tomography.

For first-order optimization methods (such as SGD and Adam), it is the gradient of the loss function rather than the loss function itself that must be estimated. From Eq.˜11 and using the log-derivative trick, we obtain the gradient

where we define the diagonal operator $D_{\lambda}$ as

A simple but important observation is that the noise enters Eq.˜15 only in the overall prefactor $\propto 1/f(\mathcal{E})$. Thus, the noise may affect the learning rate, but it will not affect the direction of the gradient. This suggests that gradient-based optimization schemes can yield an accurate neural network quantum state without any noise calibration or mitigation. This is despite the fact that this same noise generally biases the estimated infidelity (see Eq.˜11).

We now discuss a possible limitation of our approach to classical post-processing. Given $N$ classical shadows collected experimentally, and $L$ computational basis samples collected from the neural network quantum state, the number of Monte Carlo samples collected from the neural network quantum state must be $L\sim O\left(1/N^{2}f(\mathcal{E})^{2}\right)$ in order to guarantee a bounded standard error in the approximation of the gradient from Eq.˜15. Since $f(\mathcal{E})\leq 1/(1+2^{n})$, this suggests that there may be an exponential cost in performing the Monte Carlo estimations. We emphasize that this potential exponential cost in classical post-processing does not affect the required number of classical shadows from measurements. As system sizes grow significantly larger, it will eventually become hopeless to perform an exact sum over all $2^{n}$ computational basis states and the Monte Carlo average may still lead to successful convergence with only a sub-exponential number of samples in some cases (further details and an alternative approach to performing the Monte Carlo average are discussed in Appendix˜D). In our numerical simulations with six qubits (see Sec.˜III), having $2^{6}=64$ computational basis states, we have evaluated Eq.˜15 using 5000 Monte Carlo samples. With this many samples, the Monte Carlo estimation error is negligible and statistical fluctuations in the gradient are predominantly due to the finite number of classical shadows collected.

Along with the standard NSQST protocol, we also outline a modified NSQST protocol we call “NSQST with pre-training”, which combines the resources used in NNQST and the standard NSQST protocol. NSQST with pre-training aims to find a solution with a lower infidelity than either of the other protocols alone. In this protocol we train two models with disjoint sets of parameters $\lambda_{1}$ and $\lambda_{2}$. We call these models the probability amplitude model and the phase model. Figure˜2 provides a visual overview of the protocol for NSQST with pre-training.

First, the parameters $\lambda_{1}$ are optimized to produce an accurate distribution $p_{\lambda_{1}}(s)\simeq p_{\Phi}(s,B)$ from measurements performed exclusively in the computational basis [$B=(Z_{1},Z_{2},\cdots,Z_{n})$]. Note that we can efficiently evaluate the loss function, Eq.˜4, and its gradient in this case as they depend only on the probabilities and not on the phases. Next, we perform NSQST to train the model parameters $\lambda_{2}$, learning the phases $\varphi_{\lambda_{2}}(s)$. However, unlike the case of standard NSQST, to perform a Monte Carlo estimate of the gradient, Eq.˜15, here we select random samples from a set of computational basis states $s$ according to the pre-trained distribution $p_{\lambda_{1}}(s)$. Since the NSQST Monte Carlo approximations do not follow the model $\lambda_{2}$ in an on policy fashion, re-sampling in every iteration is no longer necessary. Nevertheless, we still re-sampled in our numerical experiments (described below) to reduce the sampling bias, and since the classical sampling procedure was not computationally costly in our examples.

NSQST with pre-training resembles coordinate descent optimization, with $\lambda_{1}$ and $\lambda_{2}$ being the two coordinate blocks. Optimizing $\lambda_{1}$ first and fixing it for the optimization of $\lambda_{2}$ reduces the dimension of the parameter space for the optimizers throughout the training. However, this does not guarantee convergence to a better local minimum in the loss landscape. We do not intend to demonstrate a clear advantage for NSQST with pre-training over the standard NSQST protocol, as the former uses more computational resources both experimentally (by requiring more measurements) and classically (in the form of the memory, time, and energy consumed to train the neural network quantum state). See Appendix˜C for a comparison of the number of model parameters and the number of measurements used for each of the two approaches. Another motivation for introducing this modified NSQST protocol is to provide new perspectives on the differences between learning the probability amplitudes and the phases of a target quantum state, as well as to inspire other useful hybrid protocols in the future.

Quantum chromodynamics (QCD) studies the fundamental strong interaction responsible for the nuclear force [41]. Lattice gauge theory, an important non-perturbative tool for studying QCD, discretizes spacetime into a lattice and the continuum results can be obtained through extrapolation [42]. Although lattice gauge theory has been extremely successful in QCD studies, simulations of many important physical phenomena such as real-time evolution are still out of reach due to the sign problem in current simulation techniques. Quantum computers are envisioned to overcome this barrier in lattice gauge theory-based QCD simulations and they may open the door to new discoveries in QCD [43, 44, 45, 46].

We consider a Trotterized time evolution with the gauge group SU(3) and aim to reconstruct the time-evolved quantum state after a given amount of time. To this end, we use the qubit formulation in Ref. [47] and study a single unit cell of the lattice. This corresponds to $n=6$ qubits representing three quarks (red, green, blue) and three anti-quarks (anti-red, anti-green, anti-blue) as shown in Fig.˜3a. The Trotterized time evolution starts from the initial state $\ket{\Phi_{0}}=\ket{\downarrow\downarrow\downarrow}\ket{\uparrow\uparrow\uparrow}$, which is known as the strong-coupling baryon-antibaryon state. The Hamiltonian governing the evolution is (from Ref. [47]):

where

with $\tilde{m}=am$ and $x=1/(ga)^{2}$ for a lattice spacing $a$, where $m$ is the bare quark mass and $g$ is the gauge coupling constant. We use two Trotter steps in our simulation, each for time $t=1.8$. See Appendix˜B and Ref. [47] for more details on the circuit and the physical significance of this time evolution.

Figure˜3 shows the results of simulating tomography on the time-evolved state $\ket{\Phi_{SU(3)}}$ using NNQST, NSQST, and NSQST with pre-training. Note that, although the NSQST protocols (with and without pre-training) are run for 2000 iterations, we use increments of ten iterations in the plot to provide a visual comparison with NNQST (which is run for 200 epochs due to faster convergence of $L_{\lambda}$ in optimization). For NSQST with pre-training, we display the optimization progress curve only after the probabilities $p_{\lambda_{1}}(s)$ have been pre-trained. This explains the lower initial infidelity for NSQST with pre-training. See Appendix˜C for further details on the simulation hyperparameters.

Based on Fig.˜3, NSQST and NSQST with pre-training both result in a lower final-state infidelity, relative to NNQST, and both predict the mean kinetic energy values better than NNQST. Figure˜3c further depicts the optimization progress curves of a typical trial. We see that for NNQST, the cross-entropy loss $L_{\lambda}$ quickly converges with very little fluctuations, despite the continued fluctuations of the state infidelity in the lower plot near $\mathcal{L}_{\lambda}\simeq 1$, indicating a very small overlap with the target state. On the other hand, standard NSQST and NSQST with pre-training both converge to a final state very close to the target state, despite fluctuations in the loss function caused by the finite number of classical shadows in each iteration. Moreover, we notice that NSQST with pre-training not only starts with a state of lower infidelity after pre-training, but also converges to a solution of lower infidelity than standard NSQST, with much more stable convergence in the end. One unexpected outcome is that NSQST with pre-training does not have a better kinetic energy prediction than the standard NSQST despite having lower infidelity. However, this can perhaps be an artifact of insufficient statistics given only ten trials. The predicted total energy and mass are also plotted in Fig.˜13 (Appendix˜E), where NSQST and NSQST with pre-training yield significantly better predictions of total energy but not the local observable $H_{m}$.

In Fig.˜4, the amplitudes and phases are displayed for typical final neural network quantum states for each protocol. We see that the NNQST protocol fails at learning the phase structure of the target state, despite accurately learning the probability distribution. This observation is consistent with NNQST’s convergence to a poor local minimum in the lower plot of Fig.˜3c, with the infidelity values stuck at around $1.0$. On the other hand, standard NSQST and NSQST with pre-training are both successful at learning the phase structure of the target state, while NSQST with pre-training also learns the probability distribution better.

The Heisenberg model describes magnetic systems quantum mechanically. Understanding the properties of the quantum Heisenberg model is crucial in many fields, including condensed matter physics, material science, and quantum information theory [48, 49, 50]. In this example, we perform tomography on a state that has evolved in time under the action of the one-dimensional antiferromagnetic Heisenberg (AFH) Hamiltonian. We use four Trotter time steps to approximate the time evolution.

The one-dimensional AFH model Hamiltonian is

We choose $n=6$ for our simulation and we take open boundary conditions. The 6-qubit initial state is set to the classical Néel-ordered state $\ket{\Phi_{0}}=\ket{\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow}$ and our target state occurs after evolving under the Heisenberg Hamiltonian up to time $t=0.8$. The circuit describing the Trotterized time evolution is given in Appendix˜B.

Figure˜5 shows the simulation results for performing tomography on the time-evolved AFH state using NNQST, NSQST, and NSQST with pre-training. We see that NSQST and NSQST with pre-training both reach a lower final state infidelity than NNQST in Fig.˜5a. Figure˜5b displays the mean staggered magnetization (along $x$) at each site. We observe that NSQST with pre-training results in a tighter spread of values about the exact result, relative to NNQST or NSQST across all sites. Comparing NNQST and standard NSQST, we observe that the standard NSQST protocol has significantly worse predictions at sites 3 and 4 than NNQST, with a mean more than one standard error away from the exact value, despite having a significantly lower final state infidelity. This is likely due to the fact that NNQST is trained using nearly-diagonal measurement data, providing direct access to the staggered magnetization observable of interest, whereas NSQST was trained using the infidelity loss. This result demonstrates that reaching a lower infidelity does not necessarily imply a better prediction of local observables, although we can improve the standard NSQST protocol by using more classical shadows per iteration. Finally, the typical optimization progress curves in Fig.˜5c are consistent with the statistical results shown in Fig.˜5a. For NNQST, the infidelity does not converge stably, despite the convergence of its loss function.

The probability amplitudes and phases obtained after training on a time-evolved AFH state are shown in Fig.˜6. Here, the two highest peaks in $p_{\lambda}(s)$ correspond to the two N$\acute{\text{e}}$el states $\ket{\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow}$ and $\ket{\downarrow\uparrow\downarrow\uparrow\downarrow\uparrow}$ [51]. Figure˜6 further confirms the advantage of NSQST with pre-training. Not only does NSQST with pre-training find more accurate phases, it also finds a better description of the probability distribution since the pre-training involves many measurement samples from the all-$Z$ basis (whereas NNQST splits the same number of measurement samples over multiple bases).

In this last example, we consider the tomography of a phase-shifted GHZ state. Here, our target is a 6-qubit GHZ state with a relative phase of $\frac{\pi}{2}$. A GHZ state is a maximally entangled state that is highly relevant to quantum information science due to its non-classical correlations [52]. Moreover, the GHZ state is the only $n$-qubit pure state that cannot be uniquely determined from its associated $(n-1)$-qubit reduced density matrices [53], indicating genuine multipartite entanglement.

As shown in Fig.˜7a, both NSQST and NSQST with pre-training result in a significantly lower average final state infidelity than NNQST. The optimization progress curves displayed in Fig.˜7b confirm this result, as we see that the infidelity of the NNQST state is rapidly fluctuating during training, quite distinctly from the previous two examples. Given that we have employed a widely used adaptive optimizer and that we have chosen a reasonable initial learning rate ($5\cdot 10^{-3}$), the occurrence of such a divergence is likely due to the fact that the NNQST loss function does not incorporate tomographically complete information about the target GHZ state.

So far we have only compared the performance of NNQST, NSQST, and NSQST with pre-training, but an important question is whether any of the above methods has an advantage over direct shadow estimation. In this subsection, we compare the performance of NSQST with pre-training and direct shadow estimation for a one-dimensional QCD time-evolved state from Sec.˜III.1. In addition, we perform a scalability study of the phase-shifted GHZ state with up to $40$ qubits, comparing NSQST to direct shadow estimation. To minimize the number of Clifford shadows used for training, we fix $200$ Clifford shadows as training data and do not re-sample in every iteration. In addition to the original pre-training protocol from Sec.˜II.5, we adopt the improved pre-training strategy described in Appendix˜D. A typical optimization progress curve with the improved pre-training strategy is shown in Fig.˜14 from Appendix˜E, where very few iterations are required for convergence. Once training is completed, we compare the prediction errors of NSQST with pre-training, Clifford shadows, and Pauli shadows.

As shown in Fig.˜8, we have compared the absolute prediction error of NSQST with pre-training (with and without an improved strategy) and two types of direct shadow estimation methods. For a fair comparison, we have randomly sampled the same number ($1200$) measurements for each method. For shadow reconstruction, $1200$ Clifford shadows or $1200$ Pauli shadows were used. For NSQST with pre-training, $1000$ computational-basis measurements were used for pre-training and $200$ shadows were used. In Fig.˜8a, we observe that using an improved strategy, NSQST with pre-training achieves a significantly smaller prediction error than either Pauli shadows or Clifford shadows. This is expected, as the kinetic-energy Hamiltonian $H_{\mathrm{kin}}$ in Eq.˜18 contains high-weight Pauli strings, and Pauli shadows are provably efficient at predicting only local observables [8]. On the other hand, Clifford shadows have an exponentially growing variance bound for any Pauli observables irrespective of locality (since $\operatorname{Tr}(O^{2})=2^{n}$ in Eq.˜10), which explains the large prediction error for kinetic energy. In Fig.˜8b, we report the predicted error in the fidelity to the ideal time-evolved state. The Pauli shadows yield the largest prediction error in fidelity estimation due to the exponentially growing variance bound for non-local observables. Finally, in Fig.˜8c, we apply the four methods to the problem of predicting a single Pauli string with increasing weight, where we change the identity matrix to Pauli-X at each site as the weight increases. The non-local observable of interest $\langle X...\rangle$ corresponds to the Wilson loop operator in lattice gauge theory with $\mathbb{Z}_{2}$ symmetry [54]. Since predicting high-weight observables is a hard task for both shadow protocols, the prediction error from NSQST with pre-training is much lower than for either Clifford or Pauli shadows for most of the observables. We also observe a consistent increasing prediction error for the Pauli shadows as the weight increases.

A natural question that arises is the scalability of NSQST’s advantages over direct shadow estimation. While a trained neural network quantum state closely approximating the target state has more predictive power than classical shadows alone, there is no guarantee of successful convergence during training. For instance, learning a general multi-qubit probability distribution without any prior knowledge in the pre-training step is hard [55] and would eventually require exponentially growing resources. The key to having a scalable advantage is to leverage prior knowledge of the prepared target state and to find ways to impose these known constraints in the neural network ansatz and the loss function [36, 56, 57].

As a proof of concept, we numerically study the sample complexity scaling of learning a phase-shifted GHZ state using NSQST with pre-training and using the improved strategy from Appendix˜D. With $3000$ measurements in the computational basis, $200$ Clifford shadows, and $5000$ Monte Carlo samples for each system size, we investigate the scaling of the final infidelity. As shown in Fig.˜9a, the final infidelity does not grow as the system size increases. This is expected, as the multi-qubit GHZ state is sparse in the computational basis and only a single relative phase needs to be determined. Although learning the GHZ state using a neural network ansatz is a trivial example, the sparseness property of the GHZ state is not exploited by direct shadow estimation and the collected Clifford shadows as training data would not be sample-efficient at predicting Pauli observables. In Fig.˜9b, we observe that direct shadow estimation fails to predict expectation values $\langle XX...XY\rangle$ accurately and yields only values of zero as the system size increases. The result presented in Fig.˜9 suggests that there is hope to reconstruct sufficiently sparse states with sub-exponentially growing resources using NSQST, and potentially non-sparse states as well with enough known constraints imposed. Finally, we emphasize that, as compared to the neural network quantum state ansatz trained on IC-POVMs data in Ref. [30], our chosen state ansatz explained in Sec.˜II.1 is sample-efficient at predicting Pauli string observables of arbitrary weight and fidelity to any classically tractable states.

We make a final remark on the predictive power of randomized measurements alone versus a trained variational pure state. While one may generally expect the trained pure state to inherit the features of the training data, this may not be true in specific cases for NSQST and Clifford shadows, where the trained pure state’s predictive power mainly depends on the global reconstruction error (quantum infidelity), rather than an estimator for a particular observable. Moreover, the variational training framework of NSQST is not limited to a neural network quantum state with Clifford shadows. Other variational ansatzes such as matrix product states and other randomized measurement schemes such as Hamiltonian-driven shadows should be explored with proper locality adjustments, for practical advantages such as hardware-aware measurements and scalable classical post-processing [58, 59, 60, 61].