How to Incorporate Higher-order Interactions in Analog Ising Machines

TL;DR

This paper investigates methods to incorporate higher-order interactions in analog Ising machines for solving SAT problems, finding that spin-sign-based interactions significantly improve performance and robustness across interaction orders.

Contribution

It introduces and evaluates a spin-sign-based interaction method that effectively mitigates imbalances in higher-order analog Ising machines, extending previous quadratic-focused work.

Findings

Spin-sign interactions outperform amplitude-based interactions in mitigating imbalances.

The method is effective across all interaction orders in analog IMs.

Smooth approximations enable hardware implementation.

Abstract

Ising machines (IMs) are specialized devices designed to efficiently solve combinatorial optimization problems. Among such problems, Boolean Satisfiability (SAT) is particularly relevant in industrial applications. To solve SAT problems using IMs, it is crucial to incorporate higher-order interactions. However, in analog IMs, interactions of different orders scale unevenly with the continuous spin amplitudes, introducing imbalances that can significantly degrade performance. We present a numerical comparison of methods to mitigate these imbalances, evaluating time-to-solution and success rate on Uniform Random 3-SAT instances from the SATLIB benchmark set. Our results show that the most effective approach employs spin interactions that are proportional to the signs of spins, rather than their continuous amplitudes. This generalizes our previous work, which showed that such interactions best mitigate imbalances induced by external fields in quadratic analog IMs. In this work, its advantage becomes substantially more pronounced, as it naturally mitigates imbalances across all interaction orders. We further demonstrate that smooth approximations of this method make it compatible with analog hardware. Our findings underscore the central role of spin-sign-based interactions in enabling robust and scalable analog IM dynamics.