Classical algorithms, correlation decay, and complex zeros of partition functions of quantum many-body systems

TL;DR

This paper introduces a classical algorithm for estimating the partition function of quantum many-body systems above the phase transition, linking computational hardness to physical phase transitions and analyzing correlation decay via complex zeros.

Contribution

It extends Barvinok's classical counting approach to quantum systems and characterizes phase transitions through complex zeros of the partition function.

Findings

Efficient quasi-polynomial time algorithm above the phase transition

Exponential decay of correlations for systems above the transition

Correlation decay improves to constant factor in special cases

Abstract

In this paper, we present a quasi-polynomial time classical algorithm that estimates the partition function of quantum many-body systems at temperatures above the thermal phase transition point. It is known that in the worst case, the same problem is NP-hard below this point. Together with our work, this shows that the transition in the phase of a quantum system is also accompanied by a transition in the hardness of approximation. We also show that in a system of n particles above the phase transition point, the correlation between two observables whose distance is at least log(n) decays exponentially. We can improve the factor of log(n) to a constant when the Hamiltonian has commuting terms or is on a 1D chain. The key to our results is a characterization of the phase transition and the critical behavior of the system in terms of the complex zeros of the partition function. Our work extends a seminal work of Dobrushin and Shlosman on the equivalence between the decay of correlations and the analyticity of the free energy in classical spin models. On the algorithmic side, our result extends the scope of a recent approach due to Barvinok for solving classical counting problems to quantum many-body systems.