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Programmable Optical Differential Ising Machines for Phase Transition Simulation

Wenjia Zhang, Xin Ye, Jin Min Yang, Zuyuan He

Published Oct 3, 2025• 0 citations
Publisher

Abstract

Abstract Drawing inspiration from the physical Ising model, optical Ising machines have emerged as a promising paradigm in optical computing. These machines typically utilize optical phase or amplitude modulators to represent the analog spin variables, which determine the final Hamiltonian by the combinatorial configuration of the complex optical field. However, current optical modulators exhibit relatively low modulation resolution and significant technical challenges arise when scaling up optical Ising systems to maintain consistent performance across integrated device clusters. Here we propose a programmable optical differential Ising machine that provides direct access to changes in the Hamiltonian. This system streamlines unnecessary matrix calculations, thereby reducing the requirements for the size and precision of the modulator matrix. Through numerical simulations on 60 to 100 spin MaxCut benchmark problems, we determine that the minimum required resolutions for the DAC and ADC are 6 bits and 5 bits, respectively, along with a significant tolerance of 0.3 rad for phase errors and over 10 multiplexing operations. Additionally, 100-spin experiments with a dual-channel system demonstrate its potential for phase transition simulation and solving complex optimization challenges.

DOI
10.21203/rs.3.rs-7651733/v1

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Abstract Drawing inspiration from the physical Ising model, optical Ising machines have emerged as a promising paradigm in optical computing. These machines typically utilize optical phase or amplitude modulators to represent the analog spin variables, which determine the final Hamiltonian by the combinatorial configuration of the complex optical field. However, current optical modulators exhibit relatively low modulation resolution and significant technical challenges arise when scaling up optical Ising systems to maintain consistent performance across integrated device clusters. Here we propose a programmable optical differential Ising machine that provides direct access to changes in the Hamiltonian. This system streamlines unnecessary matrix calculations, thereby reducing the requirements for the size and precision of the modulator matrix. Through numerical simulations on 60 to 100 spin MaxCut benchmark problems, we determine that the minimum required resolutions for the DAC and ADC are 6 bits and 5 bits, respectively, along with a significant tolerance of 0.3 rad for phase errors and over 10 multiplexing operations. Additionally, 100-spin experiments with a dual-channel system demonstrate its potential for phase transition simulation and solving complex optimization challenges.
Optical Ising machines provide a hardware approach for combinatorial optimization and physical-system emulation, but their scalability is limited by redundant matrix operations and analog modulation precision. Here we show a programmable optical differential Ising machine that directly evaluates Hamiltonian changes rather than the full Hamiltonian. Numerical simulations from 60 to 2000 spins show that 6-bit analog–digital conversion and 5-bit digital–analog conversion are sufficient to obtain solution quality close to high-precision digital references, with a phase-error tolerance of about 0.3–0.4 rad. Experiments on a dual-channel fiber-optic prototype reproduce phase-transition behavior in representative Ising models and solve MaxCut benchmarks through a multi-spin parallel tempering (PT) routine. The system reaches the best-known cut for the 800-spin G1 instance and 99.66% of the best-known cut for the 2000-spin G22 instance within fewer than 500 PT iterations. These results establish differential optical feedback as a route for scalable Ising-model computation. The authors develop a programmable optical differential Ising machine that directly computes Hamiltonian variations to eliminate redundant full-matrix calculations for Ising combinatorial optimization and physical system emulation. Simulations and fiber-optic prototype experiments verify that the hardware only requires 5-bit DACs and 6-bit ADCs with a phase-error tolerance of 0.3–0.4 rad, and it yields optimal MaxCut solutions for the 800-spin benchmark and 99.66% of the known optimum for the 2000-spin instance via multi-spin parallel tempering.
Abstract Optical Ising machines have emerged as a promising dynamical hardware solver for computational hard optimization problems. These Ising machines typically require an optical modulator to represent the analog spin variables of these problems. However, modern day optical modulators have a relatively low modulation resolution. We therefore investigate how the low bit-resolution of optical hardware influences the performance of this type of computing platform. Based on numerical simulations, we determine the minimum required bit-resolution of an optical Ising machine for different benchmark problems of different sizes. Our study shows that a limited bit-resolution of 8bit is sufficient for the optical modulator. We also observe that the use of a 1bit-resolution modulator significantly improves the performance of the Ising machine across all considered benchmark problems.
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.
Spatial photonic Ising machines offer a novel optical platform for optimization and spin-model simulation, but existing diffraction-based schemes rely on auxiliary spins or multiplexing to encode high-rank couplings and external fields, reducing either speed or spin count. We demonstrate an amplitude-only, rank-free spatial photonic Ising machine in which arbitrary Ising Hamiltonians are encoded as Hadamard products on aligned amplitude and binary spatial modulators and read out by a single-pixel intensity measurement. The machine directly programs fully connected 797-spin Ising models with external fields at nearly 9-bit precision and operates at a constant iteration rate of ~200 Hz. By removing zero-valued product terms, the same architecture scales to sparse problems and experimentally solves a Max-Cut instance on a 424,108-vertex Mobius ladder graph. We also observe the phase transition of the Sherrington-Kirkpatrick model, demonstrating programmable optical simulation beyond low-rank couplings. These results establish amplitude modulation as a scalable route to programmable photonic Ising machines.
Ising machines promise scaling advantages when solving highly complex combinatorial optimization problems compared to traditional computers. To fully exploit this, highly parallel physical systems must be identified. Here, we constructed and characterized an all-optical Ising machine utilizing a free-space Mach–Zehnder interferometer, which theoretically supports an unlimited number of spins and all-to-all programmable spin interactions via a matrix partitioning method. Crucially, in our proposed system, all bottleneck-producing mathematical operations like matrix-vector multiplication and nonlinear operations are performed optically in parallel, while electronic devices only need to execute simple arithmetic such as addition and storage/retrieval, ensuring fast nominal operation speeds. With periodic auto-calibration, the Ising machine has achieved over 160 h of high optical stability and accurate calculation. The machine is experimentally demonstrated with MAXCUT problems involving 1024 spins. The design itself is highly amenable to fabrication on an integrated chip-based photonics platform; a smooth trade-off exists between chip size and solution time.
Recently various physical systems have been proposed for modeling Ising spin Hamiltonians appealing to solve combinatorial optimization problems with remarkable performance. However, how to implement arbitrary spin-spin interactions is a critical and challenging problem in unconventional Ising machines. Here, we propose a general gauge transformation scheme to enable arbitrary spin-spin interactions and external magnetic fields as well, by decomposing an Ising Hamiltonian into multiple Mattis-type interactions. With this scheme, a wavelength-division multiplexing spatial photonic Ising machine (SPIM) is developed to show the programmable capability of general spin coupling interactions. We exploit the wavelength-division multiplexing SPIM to simulate three spin systems: ± J models, Sherrington-Kirkpatrick models, and only locally connected J 1 - J 2 models and observe the phase transitions. We also demonstrate the ground-state search for solving Max-Cut problem with the wavelength-division multiplexing SPIM. These results promise the realization of ultrafast-speed and high–power efficiency Boltzmann sampling to a generalized large-scale Ising model.