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Empirical Sample Complexity of Neural Network Mixed State Reconstruction

Haimeng Zhao, Giuseppe Carleo, Filippo Vicentini

TL;DR

This work analyzes the sample complexity of mixed-state reconstruction using Neural Quantum States, comparing Neural Density Operator (NDO) and POVM-NQS encodings on finite-temperature TFIM. By applying Control Variates, the authors substantially reduce gradient variance and classical overhead, enabling clearer assessment of asymptotic scaling. They find that NDO offers a quadratic advantage over POVM-NQS and classical shadows for near-pure states, but this advantage deteriorates as mixedness increases, while POVM-NQS shows no such advantage regardless of mixedness. A theoretical framework links KL divergence, trace distance, and energy/infidelity scaling, and explains observed valley phenomena; results underscore the need for more efficient encodings and robust training strategies in practical quantum-state reconstruction. The work provides practical guidance for implementing NQS-based state tomography on NISQ devices and releases open-source code for broader use.

Abstract

Quantum state reconstruction using Neural Quantum States has been proposed as a viable tool to reduce quantum shot complexity in practical applications, and its advantage over competing techniques has been shown in numerical experiments focusing mainly on the noiseless case. In this work, we numerically investigate the performance of different quantum state reconstruction techniques for mixed states: the finite-temperature Ising model. We show how to systematically reduce the quantum resource requirement of the algorithms by applying variance reduction techniques. Then, we compare the two leading neural quantum state encodings of the state, namely, the Neural Density Operator and the positive operator-valued measurement representation, and illustrate their different performance as the mixedness of the target state varies. We find that certain encodings are more efficient in different regimes of mixedness and point out the need for designing more efficient encodings in terms of both classical and quantum resources.

Empirical Sample Complexity of Neural Network Mixed State Reconstruction

TL;DR

This work analyzes the sample complexity of mixed-state reconstruction using Neural Quantum States, comparing Neural Density Operator (NDO) and POVM-NQS encodings on finite-temperature TFIM. By applying Control Variates, the authors substantially reduce gradient variance and classical overhead, enabling clearer assessment of asymptotic scaling. They find that NDO offers a quadratic advantage over POVM-NQS and classical shadows for near-pure states, but this advantage deteriorates as mixedness increases, while POVM-NQS shows no such advantage regardless of mixedness. A theoretical framework links KL divergence, trace distance, and energy/infidelity scaling, and explains observed valley phenomena; results underscore the need for more efficient encodings and robust training strategies in practical quantum-state reconstruction. The work provides practical guidance for implementing NQS-based state tomography on NISQ devices and releases open-source code for broader use.

Abstract

Quantum state reconstruction using Neural Quantum States has been proposed as a viable tool to reduce quantum shot complexity in practical applications, and its advantage over competing techniques has been shown in numerical experiments focusing mainly on the noiseless case. In this work, we numerically investigate the performance of different quantum state reconstruction techniques for mixed states: the finite-temperature Ising model. We show how to systematically reduce the quantum resource requirement of the algorithms by applying variance reduction techniques. Then, we compare the two leading neural quantum state encodings of the state, namely, the Neural Density Operator and the positive operator-valued measurement representation, and illustrate their different performance as the mixedness of the target state varies. We find that certain encodings are more efficient in different regimes of mixedness and point out the need for designing more efficient encodings in terms of both classical and quantum resources.
Paper Structure (14 sections, 1 theorem, 18 equations, 5 figures, 1 table)

This paper contains 14 sections, 1 theorem, 18 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Neural Quantum State Reconstruction
  3. Neural Density Operator
  4. POVM-NQS
  5. Variance Reduction via Control Variates
  6. Results and Discussion
  7. Simulations
  8. Scaling Behavior
  9. Theoretical Analysis
  10. Conclusions
  11. Numerical Details
  12. Theoretical Details
  13. The Valley Phenomenon
  14. Depolarized Molecular Ground-states

Key Result

Proposition 1

(Trace distance bounded by KL over all Pauli basis). For two $n$-qubit states $\rho$ and $\rho'$, let $\mathrm{TD}=\|\rho-\rho'\|_1/2$ be the trace distance and use $\sigma_i\in\{I, X, Y, Z\}$ to denote Pauli operators. Define the KL over all Pauli basis as $\mathrm{KL}=\sum_{i_1, \ldots, i_n=0}^{3}

Figures (5)

  • Figure 1: NQS reconstruction of the ground state of 3-qubit 1D open-boundary transverse field Ising model using NDO with different batch sizes $B$. KL divergence, energy error and infidelity are used as metrics for performance. Dashed and solid lines represent the results of training with and without the control variates method. The dotted line marks the $1/\sqrt{B}$ scaling. All the data points are averaged over 100 random instances.
  • Figure 2: Sample complexity dependence on the reconstruction error for 3-qubit 1D open-boundary transverse field Ising model under different inverse-temperature $\beta=10, 1, 0.1$ using POVM-NQS (top) and NDO (bottom). KL divergence, energy error, infidelity and classical infidelity are used as metrics for reconstruction error. Dashed lines represent the log-log linear regression results, with the slopes and $r^2$ values indicated in the legend. All data points are averaged over 100 random instances.
  • Figure 3: The scaling exponents of reconstruction error for 3-qubit 1D open-boundary transverse field Ising model under different inverse-temperature $\beta \in [10^{-1}, 10^1]$ with POVM-NQS (top) and NDO (bottom). KL divergence, energy error, infidelity, and classical infidelity are used as metrics for reconstruction error. The error bars are given by linear regression.
  • Figure 4: The valley phenomenon in scaling exponents of infidelity for different $\beta$. The exponents observed in NDO simulations are plotted in orange, while the ones given by random perturbations are plotted in blue. Error bars and the shaded region indicate the standard deviation over 100 random instances.
  • Figure 5: Sample complexity dependence on the reconstruction error for LiH ground-state under different depolarization $p=0, 0.01, 0.1$ using POVM-NQS (upper) and NDO (lower). KL divergence, energy error, infidelity and classical infidelity are used as metrics for reconstruction error. Dashed lines represent the log-log linear regression results, with the slopes and $r^2$ values indicated in the legend. All data points are averaged over 50 random instances.

Theorems & Definitions (1)

  • Proposition 1