How Physical Dynamics Shape the Properties of Ising Machines: Evaluating Oscillators vs. Bistable Latches as Ising Spins

Abstract

Ising machines exploit the natural dynamics of physical systems to minimize the Ising Hamiltonian and thereby address computationally hard combinatorial optimization problems. This paradigm has motivated a range of physical implementations. In the electronic domain, coupled networks of oscillators and bistable latches have emerged as two prominent realizations of Ising machines and are the focus of the present work. Despite this common abstraction, we demonstrate that differences in the underlying physical dynamics of oscillators and latches lead to fundamentally different stability properties and computational behavior of the resulting dynamical systems. Specifically, we show analytically that in Bistable Latch Ising Machines (BLIMs) all discrete Ising configurations possess identical linear stability, whereas in Oscillator Ising Machines (OIMs) the Jacobian spectrum depends explicitly on the spin configuration, enabling selective destabilization of higher-energy states. Evaluating the performance of both models on MaxCut instances of varying sizes, we find that this difference in stability structure yields consistently higher-quality solutions with OIMs. These results highlight how the characteristics of the device nonlinearity directly shape the dynamical and functional properties of Ising machine implementations.

Figures (4)

• Figure 1: Temporal evolution of the state of the node in (a) BLIM, and (b) OIM, simulated on a randomly generated graph with 15 nodes and 50 edges. Simulation parameters: BLIM: $\tau=1$, $\tau_c=12$$k=10$; $K_{\mathrm{n}}=0.05$; OIM: $K_{\mathrm{osc}}=1$; $K_s=1.5$; $K_\mathrm{n}=0.05$
• Figure 2: Comparison of maximal Jacobian eigenvalues ($\lambda_{\mathrm{max}}$) across all Ising configurations for (a) the BLIM and (b) the OIM. The same graph as in Fig. 1 is considered. In the BLIM, $\lambda_{\max}$ is identical for all configurations at fixed $\tau_c$, indicating configuration-independent linear stability. In the OIM, $\lambda_{\max}$ depends explicitly on the spin configuration, reflecting configuration-dependent stability. $k=2$ (BLIM) and $K_s=1.5$ (OIM).
• Figure 3: Comparison of MaxCut values obtained using OIM and BLIM dynamics across 150 random graphs. (a) $N=50$, (b) $N=100$, and (c) $N=150$ nodes (50 instances per size). Each point represents one graph; the red diagonal (identity line) denotes equal performance. In all tested instances, the OIM achieved a cut value greater than that of the BLIM.
• Figure 4: Influence of $\tau_c$ and $k$ on BLIM dynamics. (a)–(d) Representative time evolution of node states for increasing $\tau_c$ (with fixed $k=2$). (e)–(h) Histograms of the Ising energy of configurations obtained at equilibrium over 100 independent trials for the corresponding $\tau_c$ values. (i)–(l) Time evolution of node states for increasing gain $k$ (with fixed $\tau_c=12$). (m)–(p) Energy histograms obtained over 100 independent trials for the corresponding $k$ values.