Analyzing parametric oscillator Ising machines through the Kuramoto lens
Phys. Rev. Applied 25, 034057 – Published 17 March, 2026
DOI: https://doi.org/10.1103/61zx-gs91
Abstract
Networks of coupled nonlinear oscillators are emerging as powerful physical platforms for implementing Ising machines, yet the relationship between parametric oscillator implementations and traditional oscillator-based Ising machines remains underexplored. In this work, we develop a Kuramoto-style canonical phase description of parametric oscillator Ising machines by starting from the Stuart-Landau oscillator model, the canonical normal form near a Hopf bifurcation, and a natural reduced description for many parametric oscillator implementations such as the degenerate optical parametric oscillator. The resulting phase dynamics combine the usual phase-difference coupling observed in the standard Kuramoto model along with an intrinsic phase-sum term that is generated when conjugate coupling is considered. Moreover, our formulation helps explain why explicit second-harmonic driving is unnecessary in parametric oscillators and also reveals how quasi-steady amplitude heterogeneity scales the original strength of the spin interaction with potentially adverse impacts on the solution quality. Our work helps develop a unifying view of the oscillator-based approach to designing Ising machines.
