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Mind the Gap: Where Analog Ising Machines Lose Sight of their Optimization Objective

E. M. Hasantha Ekanayake, Arvind R. Venkatakrishnan, Francesco Bullo, Nikhil Shukla

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.

Mind the Gap: Where Analog Ising Machines Lose Sight of their Optimization Objective

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.
Paper Structure (9 sections, 4 theorems, 101 equations, 4 figures, 3 tables)

This paper contains 9 sections, 4 theorems, 101 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mind the Gap
  3. Fate of the Dynamics in the Gap
  4. Bridging the Gap
  5. Conclusion
  6. Critical Points in the Bifurcation Dynamics of the DIM
  7. Critical Thresholds in Oscillator Ising Machine (OIM)
  8. Critical Thresholds in Coherent Ising Machine (CIM)
  9. Critical Thresholds in Simulated Bifurcation Machine (SBM)

Key Result

Lemma 1

Let $(\boldsymbol{v}_i)_{i=1}^N$ be the eigenvectors of the Jacobian matrix $J_{ts}^{*}$. We establish the following properties:

Figures (4)

  • Figure 1: Comparison of dynamical system formulations for minimizing the Ising Hamiltonian and their key characteristics. Across all models, a finite parameter gap ($\Delta$) emerges in which the system evolution does not correspond to minimization of the Ising ground state in the general case. DIM: Dynamical Ising Machine; OIM: Oscillator Ising Machine; CIM: Coherent Ising Machine; SBM: Simulated Bifurcation Machine. $\phi$ represents the state variable in DIM and OIM, and $x$ denotes the state variable in SBM and CIM. The parameter $K$ denotes the coupling strength in DIM and OIM, whereas $\xi$ and $\xi_0$ represent the coupling strength in CIM and SBM, respectively. In SBM, $\Delta_i$ represents the positive detuning frequency, and $K_e$ denotes the positive Kerr coefficient.
  • Figure 2: (a) Evolution of $H(\sigma_{\max}) - H(\sigma^{\ast})$ as a function of $\alpha$ for the first five instances from the Gset benchmark. (b) Box‑plot distributions of the $\delta$ for different values of $\alpha$. Each graph is simulated for 10 independent trials.
  • Figure 3: Evolution of $S(\boldsymbol{v}_{max})$ and $S_{crit}(\boldsymbol{v}_{max})$ with $\alpha$ for Möbius ladder graphs of different sizes (a) $N=6$, (b) $N=8$, (c) $N=10$, (d) $N=12$, (e) $N=14$, and (f) $N=16$.
  • Figure 4: Box‑plot showiing distribution of $\delta$ for (a) G1, (b) G2, (c) G3, (d) G4, (e) G5 graphs. Each graph is simulated for 10 independent trials.

Theorems & Definitions (4)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Lemma 3: Critical parameters ($\alpha_c,\alpha_m$) for bipartite graphs