Breakdown of Gradient-Flow Dynamics in Oscillator Ising Machines from Harmonic Misalignment

TL;DR

This paper demonstrates that oscillator Ising machines often do not follow gradient descent dynamics due to harmonic misalignment, which introduces non-conservative interactions and challenges the energy-based interpretation.

Contribution

It identifies harmonic misalignment as a key factor causing the breakdown of gradient-flow dynamics in oscillator Ising machines and introduces a metric to quantify this effect.

Findings

Harmonic misalignment leads to non-gradient, non-conservative phase dynamics.

A normalized metric quantifies the non-gradient contributions in oscillator models.

Substantial non-gradient effects are found in ring oscillators and other hardware models.

Abstract

Oscillator Ising machines (OIMs) are often viewed as physical systems that perform gradient descent on an energy landscape encoding Ising solutions. Here, we show that this interpretation is not generic and breaks down in a broad class of oscillator implementations. We establish that gradient-flow dynamics require a harmonic-by-harmonic quadrature relation between the oscillator waveform and its phase response. Deviations from this condition, which we term harmonic misalignment, introduce even components in the pairwise interaction function, leading to non-conservative phase dynamics and precluding a gradient-flow description. We introduce a normalized metric for this non-gradient contribution and evaluate it across representative oscillator models relevant to OIMs. This metric reveals substantial non-gradient contributions in ring oscillators and across other hardware-realistic oscillator models. These findings identify harmonic misalignment as a fundamental mechanism for the breakdown of energy-based dynamics in OIMs and motivate nonequilibrium analysis and algorithms that explicitly account for and potentially exploit non-gradient behavior.