Quantum-inspired Ising machine using sparsified spin connectivity

Abstract

Combinatorial optimization problems become computationally intractable as these NP-hard problems scale. We previously proposed extraction-type majority voting logic (E-MVL), a quantum-inspired algorithm using digital logic circuits. E-MVL mimics the thermal spin dynamics of simulated annealing (SA) through controlled sparsification of spin interactions for efficient ground-state search. This study investigates the performance potential of E-MVL through systematic optimization and comprehensive benchmarking against SA. The target problem is the Sherrington-Kirkpatrick (SK) model with bimodal and Gaussian coupling distributions. Through equilibrium state analysis, we demonstrate that the sparsity control mechanism provides a consistent search of the solution space regardless of the problem's coupling distribution (bimodal, Gaussian) or size. E-MVL not only achieves the best performance among all tested algorithms-solving exact solutions up to 1600 spins where the best SA baseline is limited to 400 spins-but also provides insights that significantly improve SA's own temperature scheduling. These results establish E-MVL's dual contribution as both an efficient optimizer and a practical methodology for enhancing SA performance. Moreover, FPGA implementation achieved an approximately 6-fold faster solution speed than SA.

Figures (7)

• Figure 1: Ising model and sparsification mechanism of E-MVL. (a) 4-spin fully connected Ising model. (b) Evolution of interaction sparsification corresponding to sparsity $P_{s}(t)$ reduction during ground-state search. As the number of iterations increases, the sparsity parameter $P_{s}(t)$ (solid line) decreases, whereas the number of extracted spins $n_{i}(t)$ (dashed line) increases. (c) Temporal evolution of spin-state updates with controlled sparsification of interactions following the schedule shown in (b). Three distinct phases of the spin update process show how the number of extracted spins increases as the sparsity decreases.
• Figure 2: Time evolution of energy in a trial that obtained the optimal solution. The inset shows a magnified view of the final stage, demonstrating that continued local minima escape while achieving energy convergence.
• Figure 3: Energy distributions at equilibrium for (a) E-MVL and (b) MCMC simulations in SK-Gaussian. Both methods exhibit Boltzmann distributions, demonstrating the probabilistic nature of the energy distribution. (c) Temperature $T$ reproducing the E-MVL equilibrium distributions as a function of fixed sparsity $P_{s}$ for both SK-bimodal and SK-Gaussian, plotted on separate vertical axes.
• Figure 4: Fixed sparsity-dependent energy characteristics in E-MVL: (a) Average energy and standard deviation versus $P_{s}$ for SK-bimodal and (b) for SK-Gaussian ($N =$ 100, 625, 1600). (c) Energy distributions at $P_{s} =$ 0.1--0.4 for SK-bimodal ($N =$ 625), (d) SK-bimodal ($N =$ 1600), (e) SK-Gaussian ($N$ = 625), and (f) SK-Gaussian ($N =$ 1600) distributions. For energy distributions of $N =$ 100, the results for SK-bimodal are shown in Fig. 3(a) of our previous study yoshida2022mimicking, and those for SK-Gaussian are shown in Fig. \ref{['fig:Figure3']}(a) of this study.
• Figure 5: Normalized average energy $E/E_{GS}$ in the equilibrium state. (a) $E/E_{GS}$ is plotted as a function of the fixed sparsity parameter $P_{s}$. The plot includes data for both the SK-bimodal and SK-Gaussian models across three system sizes ($N =$ 100, 625, and 1600). All six conditions almost perfectly overlap. (b) $E/E_{GS}$ is plotted as a function of the MCMC temperature $T$ for the same problem types and system sizes. In contrast to (a), the curves exhibit clear separation depending on both the coupling distribution type and the system size.