Entanglement Features of Random Neural Network Quantum States

TL;DR

This paper analytically explores the entanglement properties of quantum states generated by random RBMs, revealing distinct phases and showing that some states mimic Haar-random entanglement but lack ergodicity and state design properties.

Contribution

It introduces an analytic framework to study random RBM-encoded quantum states, identifying different phases and their entanglement characteristics in the thermodynamic limit.

Findings

Identifies phases with near-maximal entanglement entropy.

Shows some RBM states resemble Haar-random states in entanglement.

States do not form quantum state designs and exhibit nonergodic behavior.

Abstract

Restricted Boltzmann machines (RBMs) are a class of neural networks that have been successfully employed as a variational ansatz for quantum many-body wave functions. Here, we develop an analytic method to study quantum many-body spin states encoded by random RBMs with independent and identically distributed complex Gaussian weights. By mapping the computation of ensemble-averaged quantities to statistical mechanics models, we are able to investigate the parameter space of the RBM ensemble in the thermodynamic limit. We discover qualitatively distinct wave functions by varying RBM parameters, which correspond to distinct phases in the equivalent statistical mechanics model. Notably, there is a regime in which the typical RBM states have near-maximal entanglement entropy in the thermodynamic limit, similar to that of Haar-random states. However, these states generically exhibit nonergodic behavior in the Ising basis, and do not form quantum state designs, making them distinguishable from Haar-random states.