Expressive power of complex-valued restricted Boltzmann machines for solving non-stoquastic Hamiltonians

TL;DR

This paper investigates the capabilities of complex-valued restricted Boltzmann machines in representing ground states of quantum spin chains, especially focusing on non-stoquastic Hamiltonians and their challenging phases.

Contribution

It provides a systematic analysis of RBM-based variational Monte Carlo, highlighting when it can effectively represent ground states and identifying phases that pose difficulties.

Findings

RBMs can accurately represent ground states when Hamiltonians are phase connected to stoquastic points.

Certain non-stoquastic phases are challenging for RBMs, especially with complex sign and amplitude structures.

Sampling efficiency varies across different phases, affecting the practical use of RBMs.

Abstract

Variational Monte Carlo with neural network quantum states has proven to be a promising avenue for evaluating the ground state energy of spin Hamiltonians. However, despite continuous efforts the performance of the method on frustrated Hamiltonians remains significantly worse than those on stoquastic Hamiltonians that are sign-free. We present a detailed and systematic study of restricted Boltzmann machine (RBM) based variational Monte Carlo for quantum spin chains, resolving how relevant stoquasticity is in this setting. We show that in most cases, when the Hamiltonian is phase connected with a stoquastic point, the complex RBM state can faithfully represent the ground state, and local quantities can be evaluated efficiently by sampling. On the other hand, we identify several new phases that are challenging for the RBM Ansatz, including non-topological robust non-stoquastic phases as well as stoquastic phases where sampling is nevertheless inefficient. Furthermore, we find that an accurate neural network representation of ground states in non-stoquastic phases is hindered not only by the sign structure but also by their amplitudes.