Analyzing Parametric Oscillator Ising Machines through the Kuramoto Lens

TL;DR

This paper develops a Kuramoto-style phase model for parametric oscillator Ising machines, unifying their description and explaining key features like phase coupling and amplitude effects, aiding in their design and analysis.

Contribution

It introduces a canonical phase description based on the Stuart-Landau model, linking parametric oscillators to traditional Kuramoto models and clarifying their dynamics.

Findings

The phase dynamics include both phase-difference and phase-sum coupling.

Explicit second-harmonic driving is unnecessary in these systems.

Amplitude heterogeneity affects the effective spin interaction strength.

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 (DOPO) among others. 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.