A subpolynomial-time algorithm for the free energy of one-dimensional quantum systems in the thermodynamic limit

TL;DR

This paper presents a classical subpolynomial-time algorithm for approximating the free energy of one-dimensional quantum systems at finite temperature, leveraging spectral radius computations of a transfer matrix.

Contribution

It introduces a novel, simple algorithm that runs in subpolynomial time for finite-temperature free energy approximation, improving over previous polynomial-time methods.

Findings

Algorithm runs in subpolynomial time for fixed T > 0

Spectral radius of a linear map is used for approximation

Eigenvector provides Gibbs state marginals

Abstract

We introduce a classical algorithm to approximate the free energy of local, translation-invariant, one-dimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperature $T = 0$) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed temperature $T > 0$ in subpolynomial time, i.e., in time $O((\frac{1}{\varepsilon})^{c})$ for any constant $c > 0$ where $\varepsilon$ is the additive approximation error. Previously, the best known algorithm had a runtime that is polynomial in $\frac{1}{\varepsilon}$. Our algorithm is also particularly simple as it reduces to the computation of the spectral radius of a linear map. This linear map has an interpretation as a noncommutative transfer matrix and has been studied previously to prove results on the analyticity of the free energy and the decay of correlations. We also show that the corresponding eigenvector of this map gives an approximation of the marginal of the Gibbs state and thereby allows for the computation of various thermodynamic properties of the quantum system.