Polynomial-time Classical Simulation for One-dimensional Quantum Gibbs States

Abstract: This paper discusses a classical simulation to compute the partition function (or free energy) of generic one-dimensional quantum many-body systems. Many numerical methods have previously been developed to approximately solve one-dimensional quantum systems. However, there exists no exact proof that arbitrary one-dimensional quantum Gibbs states can be efficiently solved by a classical computer. Therefore, the aim of this paper is to prove this with the clustering properties for arbitrary finite temperatures $\beta^{-1}$. We explicitly show an efficient algorithm that approximates the partition function up to an error $\epsilon$ with a computational cost that scales as $n\cdot {\rm poly}(1/\epsilon)$, where the degree of the polynomial depends on $\beta$ as $e^{O(\beta)}$. Extending the analysis to higher dimensions at high temperatures, we obtain a weaker result for the computational cost n\cdot (1/\epsilon)^{\log^{D-1} (1/\epsilon)}, where $D$ is the lattice dimension.