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SPIE Machine Learning in Photonics

Parallel processing of spatial photonic Ising machines using spatial multiplexing

Suguru Shimomura, Yusuke Ogura, Jun Tanida

Published Apr 8, 2024• 1 citation
Publisher

Abstract

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.

DOI
10.1117/12.3028713

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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.
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.
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.
In this study, we present a method for enhancing a function of the spatial photonic Ising machine by using time-division multiplexing. Processing of multiple signals in which spin interaction is encoded provides the representation of the Ising Hamiltonian which functions as an indicator to solve combinational optimization problems. To evaluate the performance of the proposed system, we calculated the Ising Hamiltonian by simulating light propagation. We confirmed that the optimal solution of a given Knapsack problem is effectively obtained by the proposed system.
In this study, we present a method for enhancing a function of the spatial photonic Ising machine by using time-division multiplexing. Processing of multiple signals in which spin interaction is encoded provides the representation of the Ising Hamiltonian which functions as an indicator to solve combinational optimization problems. To evaluate the performance of the proposed system, we calculated the Ising Hamiltonian by simulating light propagation. We confirmed that the optimal solution of a given Knapsack problem is effectively obtained by the proposed system.
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.
The spatial photonic Ising machine has achieved remarkable advancements in solving combinatorial optimization problems. However, it still remains a huge challenge to flexibly map an arbitrary problem to the Ising model. In this paper, we propose a general spatial photonic Ising machine based on the interaction matrix eigendecomposition method. The arbitrary interaction matrix can be configured in the two-dimensional Fourier transformation based spatial photonic Ising model by using values generated by matrix eigendecomposition. The error in the structural representation of the Hamiltonian decreases substantially with the growing number of eigenvalues utilized to form the Ising machine. In combination with the optimization algorithm, as low as ∼65% of the eigenvalues are required by intensity modulation to guarantee the best probability of optimal solution for a 20-vertex graph Max-cut problem, and this percentage decreases to below ∼20% for near-zero probability. The 4-spin experiments and error analysis demonstrate the Hamiltonian linear mapping and ergodic optimization. Our work provides a viable approach for spatial photonic Ising machines to solve arbitrary combinatorial optimization problems with the help of the multi-dimensional optical property.