Antiferromagnetic spatial photonic Ising machine through optoelectronic correlation computing

Spin-glass models are crucial for understanding interacting systems across diverse scientific and engineering fields. The development of spin machines offers promising solutions to NP-hard problems by efficiently searching for the ground states of Hamiltonian spin systems. Various platforms have been explored for this purpose, including trapped ions, atomic and photonic condensates, superconducting circuits, coupled parametric oscillators, lasers, integrated nanophotonic circuits, and polaritons. Recently, spatial photonic Ising machines (SPIMs) have emerged as a powerful tool, leveraging the speed and parallelism of optical signal processing to model large-scale Ising spin systems. While ferromagnetic and spin-glass models have been successfully implemented in SPIMs, the implementation of antiferromagnetic models, crucial in areas such as oxide materials and giant magnetoresistance, and relevant to combinatorial optimization problems like multiprocessor scheduling and circuit minimization, remained a challenge. This work addresses this gap by proposing an optoelectronic correlation computing approach for implementing antiferromagnetic models within the SPIM framework.

The paper extensively reviews existing literature on spin-glass models and various spin machine implementations, highlighting the limitations of previous approaches in handling antiferromagnetic interactions within SPIMs. It emphasizes the importance of antiferromagnetic Ising models in various scientific and engineering domains and the need for a scalable and efficient method to simulate them. The authors mention previous works on SPIMs, focusing on their limitations in handling antiferromagnetic interactions and the need for a new approach that leverages the advantages of optical signal processing. The review also touches upon the use of gauge transformations in related contexts.

The proposed methodology centers on optoelectronic correlation computing within the SPIM. The authors introduce a gauge transformation that encodes both spins and interaction strengths onto a single phase-only SLM. This transformation, illustrated in Figure 1a, rotates each original spin, projecting it onto the z-axis to obtain an effective spin. This approach maintains Hamiltonian invariance while simplifying the experimental setup. The transformation's equation is given as: φ_mn = π/2σ_j + (-1)^m+nα_mn, where α_mn = arccos ξ_mn. The experimental setup (Figure 1b) involves a collimated laser beam illuminating the SLM, with lens L1 performing a Fourier transform. The resulting intensity distribution I(u) is detected on the focal plane. A key aspect is the introduction of a distribution function g_c(u), distinct from previous target intensity approaches. The correlation function F = ∫I(u)g_c(u)du is used to evaluate the Mattis-type Ising Hamiltonian H = -F = -∑_j<hG(j-h)σ_j^zσ_h^z, where G(k) is the Fourier transform of g_c(u)sinc²(Wu/λf). This formulation allows for the implementation of antiferromagnetic models where J_jh < 0, requiring g_c(u) to have both positive and negative values, unlike previous methods. The number-partitioning problem is used as a test case to demonstrate the efficacy of the approach. The algorithm involves iteratively updating the spin configuration using a Markov chain Monte Carlo method, encoding the updated spins on the SLM, measuring the intensity distribution, evaluating the Hamiltonian through the correlation function, and accepting updates only if the Hamiltonian decreases. The experimental setup uses a collimated Gaussian beam (λ = 532 nm), a polarizer, an SLM (Holoeye PLUTO-NIR-011), and a CCD beam profiler (Ophir SP620) for intensity measurements. The SLM is calibrated using a two-shot method based on a generalized spatial differentiator. The process is then iterated to find the ground state. Detailed computation analysis is done to evaluate the computational efficiency and dominance of the optical process.

The experiments demonstrate successful ground-state search for a number-partitioning problem with N = 40,000 spins. Figure 3b shows the evolution of the Hamiltonian H and the magnetization m' during the iterative process, illustrating a rapid decrease in Hamiltonian initially and stabilization with fluctuations later. A portion of the final spin configurations (Figures 3c and 3d) are presented. The magnetization m' decreases by almost three orders of magnitude, validating the gauge transformation method. Figure 4 shows computing fidelity remains stable (within 6.9 × 10⁻³) even with an increase in N from 1600 to 40,000. Supplementary Note 3 shows that for N = 40,000, more than 94% of the total computation is performed optically, highlighting the ultrafast, high-throughput, and low-power characteristics of the optical computation process. The stability of the fidelity despite increased N is attributed to CCD background noise.

The results demonstrate the feasibility and scalability of the proposed method for implementing antiferromagnetic models in SPIMs. The use of the gauge transformation simplifies the experimental setup and improves accuracy. The significant reduction in magnetization and stable fidelity across a wide range of spin numbers showcase the effectiveness of the optoelectronic correlation computing approach. The dominance of optical computation in the overall process indicates that this approach can significantly improve the speed and efficiency compared to digital simulations. Future improvements could focus on using higher-speed SLMs and CCDs and implementing more sensitive CCD cameras to further enhance accuracy. The method is also applicable to other ground-state search algorithms, such as adiabatic evolution and simulated annealing.

This work presents a novel and efficient approach to implement antiferromagnetic models in SPIMs using optoelectronic correlation computing and gauge transformation. The experimental results, particularly the ground-state search for the large-scale number-partitioning problem, demonstrate the scalability and efficiency of this approach. Future research could explore the application of this methodology to other complex optimization problems and integrate advanced optical components for even higher speed and accuracy.

The main limitation is the impact of CCD background noise on the accuracy of the Hamiltonian evaluation and fidelity. Improving the signal-to-noise ratio using low-noise CCD devices or dynamic intensity adjustment is a key area for future improvement. The accuracy of the method is also affected by potential optical aberrations, though the use of the gauge transformation helps mitigate this issue to a large extent.