Locally accurate matrix product approximation to thermal states

TL;DR

This paper proves that thermal states in one-dimensional quantum systems can be approximated efficiently with matrix product states of bounded bond dimension, supporting the effectiveness of current numerical methods.

Contribution

It provides a rigorous bound on the bond dimension needed for accurate matrix product state approximations of thermal states at constant temperature.

Findings

Thermal states have a matrix product representation with bond dimension exponential in the square root of inverse temperature.

The approximation can achieve any desired local accuracy with a bond dimension independent of system size.

Supports the practical use of constant bond dimension in thermal state simulations.

Abstract

In one-dimensional quantum systems with short-range interactions, a set of leading numerical methods is based on matrix product states, whose bond dimension determines the amount of computational resources required by these methods. We prove that a thermal state at constant inverse temperature $\beta$ has a matrix product representation with bond dimension $e^{\tilde O(\sqrt{\beta\log(1/\epsilon)})}$ such that all local properties are approximated to accuracy $\epsilon$. This justifies the common practice of using a constant bond dimension in the numerical simulation of thermal properties.