Efficient algorithms for approximating quantum partition functions

Ryan L Mann,Tyler Helmuth

doi:10.1063/5.0013689

Abstract

We establish a polynomial-time approximation algorithm for partition functions of quantum spin models at high temperature. Our algorithm is based on the quantum cluster expansion of Netočný and Redig and the cluster expansion approach to designing algorithms due to Helmuth, Perkins, and Regts. Similar results have previously been obtained by related methods, and our main contribution is a simple and slightly sharper analysis for the case of pairwise interactions on bounded-degree graphs.

Highlights

Classical algorithms for approximating partition functions of quantum models that make use of cluster expansions have occurred in two recent papers
A quantum spin system is modelled by a hypergraph G = (X, E)
The cluster expansion is a powerful tool from mathematical physics that allows one to express, via power series expansions, perturbations of a well-understood reference model

Summary

INTRODUCTION

Classical algorithms for approximating partition functions of quantum models that make use of cluster expansions have occurred in two recent papers. In this paper, we provide a simple and concise exposition of how to construct such algorithms, with the intent of making the technique accessible to a wide audience. Classical algorithms for approximating partition functions of quantum models that make use of cluster expansions have occurred in two recent papers.. RP (randomized polynomial time) is not equal to NP (non-deterministic polynomial time) due to the results on the hardness of approximate counting.5,6 We remark that these results concern real values of β; similar computational complexity transitions from P (polynomial time) to BQP-hard (bounded-error quantum polynomial time) and P to #P-hard can be observed for complex values of β by using the methods of Refs. We note that a priori information on the location of zeros of the partition function can be combined with the methods of this paper to develop polynomial-time algorithms.

THE ABSTRACT

THE QUANTUM CLUSTER EXPANSION

APPROXIMATION ALGORITHM

CONCLUSION & OUTLOOK