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arXiv:2502.18918 · Optics Letters

Parallel spatial photonic Ising machine using spatial multiplexing for accelerating combinatorial optimization

Suguru Shimomura, Jun Tanida, Yusuke Ogura

Published Oct 7, 2025• 4 citations
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Abstract

A spatial photonic Ising machine (SPIM) handles large-scale combinatorial optimization problems due to optical processing with spatial parallelism. However, iterative feedback in the search for an optimal solution limits processing speed, even though the Ising Hamiltonian is computed optically. We propose a parallel spatial photonic Ising machine (pSPIM) utilizing spatial multiplexing to search for an optimal solution efficiently. By employing grating patterns and encoding multiple configurations of Ising spins in a phase distribution, several Ising Hamiltonians are computed simultaneously. We demonstrated that the optimal solutions of max-cut problems with 100 Ising spins were obtained more frequently as the number of processing units increased. In addition, combining the multicomponent model with parallel processing provides an efficient search for optimal solutions to problems represented by using interaction matrices with a rank greater than one. The spin-update strategy of pSPIM offers an effective approach for efficiently solving large-scale combinatorial optimization problems.

DOI
10.1364/ol.577570

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A spatial photonic Ising machine (SPIM) handles large-scale combinatorial optimization problems due to optical processing with spatial parallelism. However, iterative feedback in the search for an optimal solution limits processing speed, even though the Ising Hamiltonian is computed optically. We propose a parallel spatial photonic Ising machine (pSPIM) utilizing spatial multiplexing to search for an optimal solution efficiently. By employing grating patterns and encoding multiple configurations of Ising spins in a phase distribution, several Ising Hamiltonians are computed simultaneously. We demonstrated that the optimal solutions of max-cut problems with 100 Ising spins were obtained more frequently as the number of processing units increased. In addition, combining the multicomponent model with parallel processing provides an efficient search for optimal solutions to problems represented by using interaction matrices with a rank greater than one. The spin-update strategy of pSPIM offers an effective approach for efficiently solving large-scale combinatorial optimization problems.
A spatial photonic Ising machine (SPIM) solves large-scale combinatorial optimization problems by computing the Ising Hamiltonian through an optical Fourier transform. However, the ground-state search relies on annealing, in which spins are optimized stochastically and sequentially. We propose a ground-state search based on phase retrieval (PR), in which all spins are updated simultaneously and deterministically. Imposing an amplitude constraint in the Fourier plane guides the modulated phase distribution, which encodes the spin configuration, toward the optimal solution. We numerically demonstrate that the proposed scheme reaches the ground state of rank-one Ising Hamiltonians with 10410^4104 spins in a single iteration for all trials. Moreover, a radial rearrangement of the amplitudes and an appropriately designed target pattern alleviate search stagnation and improve the optimization of spin configurations. The collective and deterministic spin update by phase retrieval enables a fast ground-state search for large-scale combinatorial optimization.
Combinatorial optimization problems are challenging for classical computing architectures, motivating the development of specialized hardware like Ising machines. The Spatial Photonic Ising Machine (SPIM) and its enhanced versions have shown promise in solving such problems by mapping them to the Ising model and leveraging optical parallelism to compute Hamiltonian energies efficiently. However, the 0-1 knapsack problem, as a representative combinatorial optimization problem, has only been reported to be solved effectively in small-scale SPIM. In this work, we analyze the factors that limit the spin scalability and propose a novel hardware-aware Hamiltonian mapping scheme to overcome this limitation. Based on this enhanced SPIM, the scheme is validated through simulated annealing experiments on a 102400-spin 0-1 knapsack problem, achieving high solution accuracy. Our method not only enables efficient solutions to large-scale 0-1 knapsack problems but also offers generalizability to other combinatorial optimization problems with similar coupling configurations. This work demonstrates the potential of SPIMs as scalable and adaptable solvers for real-world optimization challenges.
Solving combinational optimization problems, which are known to be non-deterministic polynomial hard, is useful in a wide range of fields, including transportation and finance. A spatial photonic Ising machine (SPIM) enables problems with a large number of combinations to be solved by encoding the spin variables into phase [1]. However, it is difficult to handle various types of problems because the degree of freedom of the interaction between spin variables is equal to the number of spin variables. In this study, we propose a method for expanding the flexibility of the interaction between spin variables by parallel processing using space-division multiplexing. Space-division multiplexing can be used to process components of the Ising Hamiltonian simultaneously, and a wide range of Ising Hamiltonians can be represented by synthesizing the individual results.
Spatial photonic Ising machines (SPIMs) based on spatial light modulators (SLMs) have emerged as highly effective solvers for many tasks, including combinatorial optimization problems and spin-glass simulations. However, traditional SPIMs relying solely on the simulated annealing algorithm require a large number of measurement-feedback iterations to find a relatively optimal solution in complex energy landscapes, suffering from slow convergence and high time cost. Here, we propose an optical genetic-simulated annealing hybrid algorithm to accelerate the ground-state search of SPIMs. GA conducts a global coarse-grained search in the early iteration stage, while SA performs fine-grained local refinement in the late stage. Numerical simulations show that our method enables a higher solution quality of full-rank Max-Cut problems than pure GA or SA at different scales. We also experimentally demonstrate its superiority over conventional algorithms on a gauge-transformation time-division multiplexing SPIM for high-rank optimization problems under the same iteration budget. Our approach can be further developed with other advanced metaheuristic algorithms toward intelligent optical Ising computing systems.
In this study, we proposed a spatial photonic Ising machine with parallel processing. Through spatial multiplexing, the spin variables indicating multiple types of solutions were expanded in space. Individual calculations of the Ising Hamiltonian for multiple types of spin variables can be performed using the designed phase patterns. To evaluate the performance of the proposed method, the optimal solution for combinational optimization problems with 100 spin variables was determined for a certain number of iterations. Experimental results show that the frequency of the obtained optimised solution increases with the number of multiplexing steps.
The spatial photonic Ising machine (SPIM) [13D. Pierangeli et al., Large-Scale Photonic Ising Machine by Spatial Light Modulation, Phys. Rev. Lett. 122, 213902 (2019).PRLTAO0031-900710.1103/PhysRevLett.122.213902] is a promising optical architecture utilizing spatial light modulation for solving large-scale combinatorial optimization problems efficiently. The primitive version of the SPIM, however, can accommodate Ising problems with only rank-one interaction matrices. In this Letter, we propose a new computing model for the SPIM that can accommodate any Ising problem without changing its optical implementation. The proposed model is particularly efficient for Ising problems with low-rank interaction matrices, such as knapsack problems. Moreover, it acquires the learning ability of Boltzmann machines. We demonstrate that learning, classification, and sampling of the MNIST handwritten digit images are achieved efficiently using the model with low-rank interactions. Thus, the proposed model exhibits higher practical applicability to various problems of combinatorial optimization and statistical learning, without losing the scalability inherent in the SPIM architecture.