Approximation Algorithms for Complex-Valued Ising Models on Bounded Degree Graphs

TL;DR

This paper develops a deterministic polynomial-time approximation scheme for the complex-valued Ising model partition function on bounded degree graphs, ensuring non-vanishing and extending to quantum circuit amplitudes.

Contribution

It introduces a novel approximation algorithm for complex Ising models on bounded degree graphs, leveraging polynomial zero location and extending to quantum computations.

Findings

Approximation scheme works efficiently for parameters close to zero.

Partition function does not vanish in the considered parameter regime.

Algorithm extends to approximate quantum circuit output amplitudes.

Abstract

We study the problem of approximating the Ising model partition function with complex parameters on bounded degree graphs. We establish a deterministic polynomial-time approximation scheme for the partition function when the interactions and external fields are absolutely bounded close to zero. Furthermore, we prove that for this class of Ising models the partition function does not vanish. Our algorithm is based on an approach due to Barvinok for approximating evaluations of a polynomial based on the location of the complex zeros and a technique due to Patel and Regts for efficiently computing the leading coefficients of graph polynomials on bounded degree graphs. Finally, we show how our algorithm can be extended to approximate certain output probability amplitudes of quantum circuits.