Polynomial-Time Approximation of Zero-Free Partition Functions

TL;DR

This paper introduces a polynomial-time algorithm for approximating classical and quantum partition functions under zero-free conditions, extending previous frameworks to broader classes of models with bounded degree.

Contribution

It develops a new abstract framework that generalizes Patel and Regts' approach, enabling polynomial-time approximation of partition functions for local Hamiltonians with zero-free properties.

Findings

Provides polynomial-time approximation algorithms for classical and quantum partition functions.

Extends zero-free based algorithms to local Hamiltonians with bounded degree.

Achieves approximation when inverse temperature is sufficiently close to zero.

Abstract

Zero-free based algorithm is a major technique for deterministic approximate counting. In Barvinok's original framework[Bar17], by calculating truncated Taylor expansions, a quasi-polynomial time algorithm was given for estimating zero-free partition functions. Patel and Regts[PR17] later gave a refinement of Barvinok's framework, which gave a polynomial-time algorithm for a class of zero-free graph polynomials that can be expressed as counting induced subgraphs in bounded-degree graphs. In this paper, we give a polynomial-time algorithm for estimating classical and quantum partition functions specified by local Hamiltonians with bounded maximum degree, assuming a zero-free property for the temperature. Consequently, when the inverse temperature is close enough to zero by a constant gap, we have polynomial-time approximation algorithm for all such partition functions. Our result is based on a new abstract framework that extends and generalizes the approach of Patel and Regts.