Bootstrapping Classical Shadows for Neural Quantum State Tomography

TL;DR

This paper introduces autoregressive neural quantum states and a novel loss function for classical shadow tomography, enabling stable reconstruction of complex quantum states and improving predictions of high-weight and nonlinear observables.

Contribution

It proposes a new estimator and importance sampling strategy for training neural quantum states with classical shadows, enhancing predictive accuracy and stability.

Findings

Successful reconstruction of GHZ states using transformer-based neural networks

Enhanced prediction of high-weight and nonlinear observables

Overcoming limitations of Pauli-based classical shadow tomography

Abstract

We investigate the advantages of using autoregressive neural quantum states as ansatze for classical shadow tomography to improve its predictive power. We introduce a novel estimator for optimizing the cross-entropy loss function using classical shadows, and a new importance sampling strategy for estimating the loss gradient during training using stabilizer samples collected from classical shadows. We show that this loss function can be used to achieve stable reconstruction of GHZ states using a transformer-based neural network trained on classical shadow measurements. This loss function also enables the training of neural quantum states representing purifications of mixed states. Our results show that the intrinsic capability of autoregressive models in representing physically well-defined density matrices allows us to overcome the weakness of Pauli-based classical shadow tomography in predicting both high-weight observables and nonlinear observables such as the purity of pure and mixed states.