Mind the Gap: Where Analog Ising Machines Cease to Minimize the Ising Hamiltonian

TL;DR

This paper reveals a fundamental structural limitation in analog Ising machines caused by a gap between destabilization and stabilization phases, impacting their ability to reliably minimize the Ising Hamiltonian.

Contribution

It identifies a universal structural feature in analog Ising machines and proposes a hybrid dynamical framework to modulate the critical parameter gap.

Findings

The parameter gap prevents guaranteed convergence to Ising solutions.

Spectral analysis of the Jacobian explains the gap's influence.

A hybrid framework can reshape bifurcation topology to improve performance.

Abstract

The design of nonlinear dynamical systems whose gradient flows minimize the Ising Hamiltonian has emerged as a compelling paradigm for realizing Ising machines, forming the foundation of architectures including coherent Ising machines, simulated bifurcation machines, oscillator-based Ising machines, and dynamical Ising machines. Here, we identify a fundamental structural feature shared by these systems, a functional gap defined by the separation between the destabilization of the trivial state and the stabilization of Ising-encoded states. We demonstrate that this separation creates a finite parameter interval in which convergence to an Ising-encoded solution is no longer functionally guaranteed, and the resulting evolution is dictated by the spectral structure of the Jacobian at bifurcation. Subsequently, by introducing a hybrid dynamical framework that reshapes the bifurcation topology, we establish a principled pathway for modulating this parameter gap. The parameter gap thus emerges as a unifying structural principle for the analysis, design and optimization of analog Ising machines.