Variational neural and tensor network approximations of thermal states

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 Ansätze 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.