The complexity of approximating the complex-valued Ising model on bounded degree graphs

TL;DR

This paper investigates the computational complexity of approximating the Ising model's partition function with complex edge interactions on bounded degree graphs, establishing new tractability and intractability boundaries.

Contribution

It introduces new zero-free regions for the Ising model and proves -hardness of approximation for complex parameters outside these regions.

Findings

FPTAS exists when rac{ ext{|}eta - 1 ext{|}}{ ext{|}eta + 1 ext{|}} < an(rac{\u03c0}{4\u00a0 ext{(} ext{4}\u00a0 ext{d}d-4 ext{)}})

It is -hard to approximate the partition function when eta is complex and outside the zero-free region.

Zeros of the partition function imply hardness of approximation in certain complex parameter regimes.

Abstract

We study the complexity of approximating the partition function $Z_{\mathrm{Ising}}(G; \beta)$ of the Ising model in terms of the relation between the edge interaction $\beta$ and a parameter $\Delta$ which is an upper bound on the maximum degree of the input graph $G$. Following recent trends in both statistical physics and algorithmic research, we allow the edge interaction $\beta$ to be any complex number. Many recent partition function results focus on complex parameters, both because of physical relevance and because of the key role of the complex case in delineating the tractability/intractability phase transition of the approximation problem. In this work we establish both new tractability results and new intractability results. Our tractability results show that $Z_{\mathrm{Ising}}(-; \beta)$ has an FPTAS when $\lvert \beta - 1 \rvert / \lvert \beta + 1 \rvert < \tan(\pi / (4 \Delta - 4))$. The core of the proof is showing that there are no inputs~$G$ that make the partition function $0$ when $\beta$ is in this range. Our result significantly extends the known zero-free region of the Ising model (and hence the known approximation results). Our intractability results show that it is $\mathrm{\#P}$-hard to multiplicatively approximate the norm and to additively approximate the argument of $Z_{\mathrm{Ising}}(-; \beta)$ when $\beta \in \mathbb{C}$ is an algebraic number such that $\beta \not \in \mathbb{R} \cup \{i,-i\}$ and $\lvert \beta - 1\rvert / \lvert \beta + 1 \rvert > 1 / \sqrt{\Delta - 1}$. These are the first results to show intractability of approximating $Z_{\mathrm{Ising}}(-, \beta)$ on bounded degree graphs with complex $\beta$. Moreover, we demonstrate situations in which zeros of the partition function imply hardness of approximation in the Ising model.