Ising Model Partition Function Computation as a Weighted Counting Problem

TL;DR

This paper explores the computational complexity of the Ising model's partition function, relating it to #CSP problems, and demonstrates how reducing it to Weighted Model Counting enables more efficient computation using existing tools.

Contribution

It establishes the hardness of #Ising via #CSP dichotomies and introduces a reduction to Weighted Model Counting, improving computational methods for physics problems.

Findings

#Ising is computationally hard, as shown by #CSP results.

Reducing #Ising to WMC allows use of existing model counters.

WMC-based approach outperforms specialized #Ising solvers.

Abstract

While the Ising model remains essential to understand physical phenomena, its natural connection to combinatorial reasoning makes it also one of the best models to probe complex systems in science and engineering. We bring a computational lens to the study of Ising models, where our computer-science perspective is two-fold: On the one hand, we consider the computational complexity of the Ising partition-function problem, or #Ising, and relate it to the logic-based counting of constraint-satisfaction problems, or #CSP. We show that known dichotomy results for #CSP give an easy proof of the hardness of #Ising and provide new intuition on where the difficulty of #Ising comes from. On the other hand, we also show that #Ising can be reduced to Weighted Model Counting (WMC). This enables us to take off-the-shelf model counters and apply them to #Ising. We show that this WMC approach outperforms state-of-the-art specialized tools for #Ising, thereby expanding the range of solvable problems in computational physics.