Variational neural and tensor network approximations of thermal states

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 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 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>N</a:mi><a:mo>=</a:mo><a:mn>100</a:mn></a:mrow></a:math> 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.