How to Incorporate External Fields in Analog Ising Machines

Abstract: Ising machines (IMs) are specialized devices designed to efficiently solve combinatorial optimization problems (COPs). They consist of artificial spins that evolve towards a low-energy configuration representing a problem's solution. Most realistic COPs require both spin-spin couplings and external fields. In IMs with analog spins, these interactions scale differently with the continuous spin amplitudes, leading to imbalances that affect performance. Various techniques have been proposed to mitigate this issue, but their performance has not been benchmarked. We address this gap through a numerical analysis. We evaluate the time-to-solution of these methods across three distinct problem classes with up to 500 spins. Our results show that the most effective way to incorporate external fields is through an approach where the spin interactions are proportional to the spin signs, rather than their continuous amplitudes.