Global Optimization Through Heterogeneous Oscillator Ising Machines

TL;DR

This work develops a signed-graph Laplacian framework to analyze oscillator Ising machines (OIMs) for solving Ising minimization problems, linking the energy landscape to the Hessian spectrum of the OIM energy. It shows that in frustration-free networks a suitable positive regularization $\\mu$ guarantees asymptotic stability of global minimizers, while in frustrated ensembles low-energy configurations are statistically more likely to be stable. Importantly, introducing heterogeneity in the regularization parameters $\\mu_i$ increases the Hessian variance at extreme energies, biasing dynamics toward globally optimal equilibria with a larger spectral gap. Numerical experiments corroborate the theory and motivate design guidelines that use parameter heterogeneity to achieve high-probability convergence to global optima.

Abstract

Oscillator Ising machines (OIMs) are networks of coupled oscillators that seek the minimum energy state of an Ising model. Since many NP-hard problems are equivalent to the minimization of an Ising Hamiltonian, OIMs have emerged as a promising computing paradigm for solving complex optimization problems that are intractable on existing digital computers. However, their performance is sensitive to the choice of tunable parameters, and convergence guarantees are mostly lacking. Here, we show that lower energy states are more likely to be stable, and that convergence to the global minimizer is often improved by introducing random heterogeneities in the regularization parameters. Our analysis relates the stability properties of Ising configurations to the spectral properties of a signed graph Laplacian. By examining the spectra of random ensembles of these graphs, we show that the probability of an equilibrium being asymptotically stable depends inversely on the value of the Ising Hamiltonian, biasing the system toward low-energy states. Our numerical results confirm our findings and demonstrate that heterogeneously designed OIMs efficiently converge to globally optimal solutions with high probability.

Figures (3)

• Figure 1: Example of a frustrated network with antiferromagnetic interactions between nodes (depicted by dashed lines), where for any spin configuration there exists a pair of adjacent nodes with aligned spins. (a) Triangular spin interaction network where spins 1 and 2 are antialigned, thereby minimizing their interaction energy. (b) State in which spin 3 is antialigned with spin 2 and aligned with spin 1; consequently not minimizing the local interaction energy $-J_{13}\sigma_1\sigma_3$. (c) Complementary state in which the local interaction energy between spins 3 and 2, $-J_{23}\sigma_2\sigma_3$, is not minimized.
• Figure 2: Signed graph construction. (a) Graph of the Ising model and corresponding spin configuration $\sigma$, where ferromagnetic and antiferromagnectic edges are denoted by solid and dashed lines, respectively. (b) Signed graph $G_{\rm s}(\sigma)$, where blue and red lines denote positive and negative edges, respectively.
• Figure 3: Statistical analysis of the Ising energy and the stability of OIMs. (a, b) Distributions of the Ising Hamiltonian energy $\mathcal{H}$ (a) and Hessian eigenvalues $\lambda$ (b) across $10^5$ realizations of signed graphs from the ensemble $\mathcal{G}$. The horizontal bars indicate the mean $\pm$ standard deviation, with blue and red colors corresponding to theoretical predictions and numerical estimates, respectively. (c, d, e) Conditional moments for given values of the Ising Hamiltonian energy. The theoretical predictions are shown by blue lines, while black data points represent numerical estimates. In panel c, green points represent numerical estimates of the conditional expectation of the smallest Hessian eigenvalue (i.e., $\mathbb{E}[\lambda_{\rm min} | \mathcal{H}]$). In panels d and e, the yellow line shows a second-order polynomial fit for comparison. In all plots, the statistical results were obtained for graphs of $50$ nodes, with the probabilities $p_1$ and $p_2$, and the heterogeneity interval $[a,b]$ reported in the corresponding columns.

Theorems & Definitions (12)

• Example 1
• Theorem 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 2