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arXiv:2609.16474

Deterministic Ground-State Search in a Spatial Photonic Ising Machine by Phase Retrieval

Suguru Shimomura, Jun Tanida, Yusuke Ogura

Published Sep 15, 2026• 0 citations
arXivHTMLPDF

Abstract

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 $10^4$ 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.

DOI
10.48550/arxiv.2609.16474

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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.
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.
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.
The spatial photonic Ising machine (SPIM) is a simple and elegant architecture capable of expanding the spin scale with ease, and thus this optical architecture is attractive for its potential in solving large-scale combinatorial optimization problems. Here, we propose a scheme for aberration compensation to improve the computing accuracy of the SPIM based on a spatial light modulator (SLM), resulting in good scalability. First, the gamma curve of the SLM is corrected with the help of double-hole interference fringes. Then, an enhanced phase shift interferometry is used to measure the aberration of the entire optical system, and this aberration is compensated by loading the compensation phase on the SLM. Moreover, an iterative algorithm using self-adaptive cluster flipping is proposed for accelerating the convergence of SPIM. During the converging process, the number of flipped spins within the cluster is updated dynamically according to the convergence speed and accuracy, which makes it possible to achieve a good balance between high speed and high accuracy. Further, a proof-of-principle experimental set-up of the optimized SPIM is implemented, and the ground-state search of an antiferromagnetic Ising model is demonstrated. As an example, this optimized Ising machine is used for solving the number partitioning problem. It is shown that the partition of 40,000 random numbers is well demonstrated, and the final partitioning accuracy is as high as 99.88%.
Photonic Ising Machines constitute an emergent new paradigm of computation, geared towards tackling combinatorial optimization problems that can be reduced to the problem of finding the ground state of an Ising model. Spatial Photonic Ising Machines have proven to be advantageous for simulating fully connected large-scale spin systems. However, fine control of a general interaction matrix JJJ has so far only been accomplished through eigenvalue decomposition methods that either limit the scalability or increase the execution time of the optimization process. We introduce and experimentally validate a SPIM instance that enables direct control over the full interaction matrix, enabling the encoding of Ising Hamiltonians with arbitrary couplings and connectivity. We demonstrate the conformity of the experimentally measured Ising energy with the theoretically expected values and then proceed to solve both the unweighted and weighted graph partitioning problems, showcasing a systematic convergence to an optimal solution via simulated annealing. Our approach greatly expands the applicability of SPIMs for real-world applications without sacrificing any of the inherent advantages of the system, and paves the way to encoding the full range of NP problems that are known to be equivalent to Ising models, on SPIM devices.
Photonic Ising machines constitute an emergent new paradigm of computation geared toward tackling combinatorial optimization problems that can be reduced to the problem of finding the ground state of an Ising model. Spatial photonic Ising machines (SPIMs) have proven advantageous for simulating fully connected large-scale spin systems. Fine control of a general interaction matrix J has been accomplished so far only through matrix decomposition methods. We introduce and experimentally validate a SPIM instance that enables direct control over the full interaction matrix, allowing the encoding of Ising Hamiltonians with arbitrary couplings and connectivity. We demonstrate the conformity of the experimentally measured Ising energy with the theoretically expected value and then proceed to solve both the unweighted and weighted graph partitioning problems, showcasing a systematic convergence to an optimal solution via simulated annealing. Our approach significantly expands the applicability of SPIMs for real-world applications, as it is more efficient than matrix decomposition methods in the case of sparse problems. It paves the way to encoding the full range of NP problems that are known to be equivalent to Ising models on SPIM devices.
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.