Ising model partition-function computation as a weighted counting problem.

Abstract

While the Ising model is most often used to understand physical phenomena, its natural connection to combinatorial reasoning also makes it 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 twofold: On the one hand, we show that partition function computation (#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 one model counter (TensorOrder) outperforms state-of-the-art tools for #Ising on midsize and topologically unstructured instances, suggesting the tool would be a useful addition to a portfolio of partition function solvers. On the other hand, we consider the computational complexity of #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 intuition on where the difficulty of #Ising comes from.