# Transforming Generalized Ising Model into Boltzmann Machine

## TL;DR

This paper presents an exact method to convert generalized Ising models with many-spin interactions into equivalent Boltzmann machines with only two-spin interactions, improving simulation efficiency.

## Contribution

The authors develop a precise algebraic mapping using star-triangle and decoration-iteration transformations, enabling simpler models and enhanced Monte Carlo simulations.

## Key findings

- Mapping reduces model complexity to two-spin interactions.
- Application with Swendsen-Wang algorithm decreases critical slowing down.
- Effective in models with multi-spin interactions on Kagomé lattice.

## Abstract

We find an exact mapping from the generalized Ising models with many-spin interactions to equivalent Boltzmann machines, i.e., the models with only two-spin interactions between physical and auxiliary binary variables accompanied by local external fields. More precisely, the appropriate combination of the algebraic transformations, namely the star-triangle and decoration-iteration transformations, allows one to express the model in terms of fewer-spin interactions at the expense of the degrees of freedom. Furthermore, the benefit of the mapping in Monte Carlo simulations is discussed. In particular, we demonstrate that the application of the method in conjunction with the Swendsen-Wang algorithm drastically reduces the critical slowing down in a model with two- and three-spin interactions on the Kagom\'e lattice.