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

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.

Automated summary

We present a classical algorithm for approximating the free energy of one-dimensional quantum systems with local, translation-invariant properties in the thermodynamic limit. While solving the ground state problem for these systems is computationally difficult, our algorithm provides a subpolynomial time solution for fixed temperatures above absolute zero. The algorithm computes the spectral radius of a linear map, which is interpreted as a noncommutative transfer matrix and has been studied in relation to the analyticity of the free energy and correlation decay. The corresponding eigenvector of this map allows for the computation of various thermodynamic properties of the quantum system.