Variational Neural and Tensor Network Approximations of Thermal States

TL;DR

This paper presents a variational Monte Carlo method for approximating finite-temperature quantum many-body states using neural and tensor network ansatzes, enabling systematic improvements without iterative errors.

Contribution

Introduces a novel variational Monte Carlo approach for finite-temperature states that directly optimizes the free energy using neural and tensor network states.

Findings

Tensor network states outperform neural networks in accuracy.

Systematic improvement observed with increased bond dimension.

Method applicable to systems with up to 100 spins.

Abstract

We introduce a variational Monte Carlo algorithm for approximating finite-temperature quantum many-body systems, based on the minimization of a modified free energy. This approach directly approximates the state at a fixed temperature, allowing for systematic improvement of the ansatz expressiveness without accumulating errors from iterative imaginary time evolution. We employ a variety of trial states -- both tensor networks as well as neural networks -- as variational Ans\"atze for our numerical optimization. We benchmark and compare different constructions in the above classes, both for one- and two-dimensional problems, with systems made of up to $N=100$ spins. Our results demonstrate that while restricted Boltzmann machines show limitations, string bond tensor network states exhibit systematic improvements with increasing bond dimensions and the number of strings.