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Cambridge University Press eBooks

The Spatial Ising Machine

Claudio Conti

Published Jun 26, 2026• 0 citations
Publisher

Abstract

We show how spatial light modulation allows one to optically compute an Ising Hamiltonian. This enables the use of optical radiation to accelerate algorithms to find the ground state and hence the optimal solution of a combinatorial optimization problem. The phase matrix on a spatial light modulator acts as a lattice of spins whose interaction is ruled by the constrained optical intensity in the far field. Feedback from the detection plane allows the spatial phase distribution to evolve toward the minimum of the selected spin model. Related topics such as annealing, adiabatic evolution, the XY model, and decomposition methods are reviewed.

DOI
10.1017/9781316275207.015

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We show how spatial light modulation allows one to optically compute an Ising Hamiltonian. This enables the use of optical radiation to accelerate algorithms to find the ground state and hence the optimal solution of a combinatorial optimization problem. The phase matrix on a spatial light modulator acts as a lattice of spins whose interaction is ruled by the constrained optical intensity in the far field. Feedback from the detection plane allows the spatial phase distribution to evolve toward the minimum of the selected spin model. Related topics such as annealing, adiabatic evolution, the XY model, and decomposition methods are reviewed.
Combinatorial optimization problems are crucial for widespread applications but remain difficult to solve on a large scale with conventional hardware. Novel optical platforms, known as coherent or photonic Ising machines, are attracting considerable attention as accelerators on optimization tasks formulable as Ising models. Annealing is a well-known technique based on adiabatic evolution for finding optimal solutions in classical and quantum systems made by atoms, electrons, or photons. Although various Ising machines employ annealing in some form, adiabatic computing on optical settings has been only partially investigated. Here, we realize the adiabatic evolution of frustrated Ising models with 100 spins programmed by spatial light modulation. We use holographic and optical control to change the spin couplings adiabatically, and exploit experimental noise to explore the energy landscape. Annealing enhances the convergence to the Ising ground state and allows to find the problem solution with probability close to unity. Our results demonstrate a photonic scheme for combinatorial optimization in analogy with adiabatic quantum algorithms and classical annealing methods but enforced by optical vector-matrix multiplications and scalable photonic technology.
Many problems in mathematics, statistical mechanics, and computer science are hard to tackle by conventional computational methods, but can be reduced to the combinatorial problem of finding the ground state of the Ising model. However, the computational cost associated with a combinatorial optimization problem increases exponentially with the number of variables. This explains growing research interest towards physical and artificial spin-networks (schematically shown in Fig. 1 (a) ) that evolve according to an Ising Hamiltonian, which provides the combinatorial solution through the ground state observed in the experiment [1] . The implementation of analog autonomous spin-networks is currently considered a highly attractive prospect due to a potentially significant increase in energy efficiency and processing speed.
With the growing demand for efficient computing to tackle complex real-world challenges like combinatorial optimization, both academia and industry are exploring alternative computing paradigms. The Ising machine (IM) is one such approach, solving optimization problems by mapping their cost functions onto the Ising model. In this framework, finding the optimal solution equates to determining the Ising spin configuration corresponding to the ground state of the Ising Hamiltonian. Analog IMs, relying on artificial spins that can take analogue rather than binary values, have shown great potential in reaching this ground at high speeds. Optical implementations, such as the opto-electronic IM developed within our research group, are particularly promising due to their higher bandwidth, energy efficiency, and lower latency compared to electronic IMs. However, scaling remains a challenge for large problems. To overcome this, we propose a novel analog optical IM utilizing a spatial light modulator (SLM). Our design differs from other IMs relying on SLMs by employing analog spin variables instead of binary ones and leveraging gradient descent-based feedback rather than a computer-driven Metropolis–Hastings sampling process. By exploiting the inherent parallelism of light, this SLM-based IM is expected to efficiently handle optimization problems with millions of spins while still maintaining high energy efficiency. At the conference, we will also offer more insight into the performance of these analogue IMs and will delve deeper in the bit resolution required for the different optical components such as the employed SLM. Based on the dynamical evolution of the IM, we will discuss strategies to further improve the problem-solving capacity and speed of these optical IMs.
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.