Stability of Oscillator Ising Machines: Not All Solutions Are Created Equal

TL;DR

This paper analyzes the stability of oscillator-based Ising machines using nonlinear control theory, revealing that not all optimal solutions are equally stable, which affects the quality and bias of solutions.

Contribution

It introduces a nonlinear control-theoretic framework to assess stability in oscillator Ising machines, highlighting that some globally optimal solutions may be unstable and biased.

Findings

Not all global minima are stable in oscillator Ising machines.

Stability of local optima influences solution quality.

Biased solutions emerge due to stability differences.

Abstract

Nonlinear dynamical systems such as coupled oscillators are being actively investigated as Ising machines for solving computationally hard problems in combinatorial optimization. Prior works have established the equivalence between the global minima of the Lyapunov function (commonly referred to as the energy function) describing the coupled oscillator system and the ground state of the Ising Hamiltonian. However, the properties of the oscillator Ising machine (OIM) from a nonlinear control viewpoint, such as the stability of the OIM solutions remains unexplored. Therefore, in this work, using nonlinear control-theoretic analysis, we (i) Identify the conditions required to ensure the functionality of the coupled oscillators as an Ising machine; (ii) Show that all globally optimal phase configurations may not always be stable, resulting in some configurations being more favored over others, and thus, creating a biased OIM; (c) Elucidate the impact of the stability of locally optimal phase configurations on the quality of the solution computed by the system. Our work, fostered through the unique convergence between nonlinear control theory and analog systems for computing, provides a new toolbox for the design and implementation of dynamical system-based computing platforms.