When can classical neural networks represent quantum states?

TL;DR

This paper investigates the conditions under which classical neural networks can efficiently represent quantum states, emphasizing the role of measurement-induced correlations and basis dependence in their expressive power.

Contribution

It introduces a theoretical framework linking measurement-induced correlations and basis choice to the neural network's ability to represent quantum states.

Findings

Conditional correlations influence neural network representations.

Measurement basis affects the correlation patterns and expressiveness.

Entanglement and sign structure determine the correlation range.

Abstract

A naive classical representation of an n-qubit state requires specifying exponentially many amplitudes in the computational basis. Past works have demonstrated that classical neural networks can succinctly express these amplitudes for many physically relevant states, leading to computationally powerful representations known as neural quantum states. What underpins the efficacy of such representations? We show that conditional correlations present in the measurement distribution of quantum states control the performance of their neural representations. Such conditional correlations are basis dependent, arise due to measurement-induced entanglement, and reveal features not accessible through conventional few-body correlations often examined in studies of phases of matter. By combining theoretical and numerical analysis, we demonstrate how the state's entanglement and sign structure, along with the choice of measurement basis, give rise to distinct patterns of short- or long-range conditional correlations. Our findings provide a rigorous framework for exploring the expressive power of neural quantum states.