General Oscillator-Based Ising Machine Models with Phase-Amplitude Dynamics and Polynomial Interactions

Abstract

We present an oscillator model with both phase and amplitude dynamics for oscillator-based Ising machines (OIMs). The model targets combinatorial optimization problems with polynomial cost functions of arbitrary order and addresses fundamental limitations of previous OIM models through a mathematically rigorous formulation with a well-defined energy function and corresponding dynamics. The model demonstrates monotonic energy decrease and reliable convergence to low-energy states. Empirical evaluations on 3-SAT problems show significant performance improvements over existing phase-amplitude models. Additionally, we propose a flexible, generalizable framework for designing higher-order oscillator interactions, from which we derive a practical method for oscillator binarization without compromising performance. This work strengthens both the theoretical foundation and practical applicability of oscillator-based Ising machines for complex optimization problems.

Figures (5)

• Figure 1: Percentage of solvable instances across problem sets (UF20-91 to UF150-645) from SATLIB hoos_satlib_2000 under 100 runs per instance. Instances are classified as solvable if at least one run reaches the ground state ($\mathcal{H}=0$). The error bars denote the 99% bootstrap confidence intervals (computed over 10,000 resamples). The proposed Hopf model consistently outperforms the baseline model bybee_efficient_2023.
• Figure 2: Final energy distribution for solving 3-SAT problems across approximately 170,000 runs per model. The proposed Hopf model achieves lower final energies with higher frequency, as shown in the histogram (top). The cumulative plot (bottom) further demonstrates the improved performance, showing a higher percentage of trials for the proposed model successfully reaching optimal or near-optimal solutions.
• Figure 3: Example phase plots, where a system can either be strongly/weakly binarized (which corresponds the variance of the phase distribution) and correctly/incorrectly binarized (which corresponds to classification errors from axis misalignment).
• Figure 4: Polar histograms of phase values for the different binarization strategies discussed.
• Figure 5: Cumulative distribution of final energy values for the different binarization strategies discussed. Each line represents the distribution of 45 problems from uf100-430 with 100 trials each. 99% confidence intervals are provided as shaded lines around each estimate, however they are typically too narrow to be visible.