Antiferromagnetic spatial photonic Ising machine through optoelectronic correlation computing

The study investigates how to realize antiferromagnetic Ising models within spatial photonic Ising machines (SPIM). Spin-glass models are central to understanding interacting systems across physics and engineering, and physical spin machines offer prospects for solving NP-hard optimization problems by searching Hamiltonian ground states. While SPIMs have demonstrated ferromagnetic and spin-glass models with thousands of spins, implementing antiferromagnetic models remained open. Antiferromagnetic Ising models are important in materials science (e.g., oxide materials, giant magnetoresistance) and mapping to combinatorial optimization problems with real-world applications (scheduling, circuit design, cryptography, logistics). The research proposes an optoelectronic correlation computing approach, leveraging gauge transformation to encode spins and interaction strengths on a single phase-only SLM, enabling evaluation of antiferromagnetic Hamiltonians and accelerating ground-state search for large-scale instances such as the number-partitioning problem.

The paper situates SPIM within a broader landscape of spin-based optimization and analog computing platforms, citing implementations using trapped ions, condensates, superconducting circuits, parametric oscillators, lasers, integrated photonics, and polaritons. Prior SPIM work achieved large-scale spin-glass and ferromagnetic models with spatial light modulators and target-intensity approaches, offering high-speed, parallel optical processing. However, previous SPIM methods could not directly implement antiferromagnetic interactions due to the non-negativity constraint of target intensities. The authors draw on earlier work on gauge transformations in SPIM to avoid pixel alignment issues and improve stability and fidelity, and on advances in optical analog computation (spatial differentiators, metamaterials) that exploit spatial degrees of freedom.

The authors develop an optoelectronic correlation computing framework for a Mattis-type Ising model H = −Σ_{j<h} J_{jh} σ_j σ_h with J_{jh} = G(j−h). They apply a gauge transformation that rotates each spin by α_j = arccos ξ_j to define effective z-components σ_j^z = ξ_j σ_j, leaving the Hamiltonian invariant: H = −Σ_{j<h} G(j−h) σ_j^z σ_h^z. This permits encoding spins and interaction strengths on a single phase-only SLM with macropixels on a 2D square lattice j = (m,n). Each macropixel phase is φ_{mn} = (π/2) σ_j + (−1)^{m+n} α_{mn}, with α_{mn} = arccos ξ_{mn}. A lens L1 (focal length f) performs a Fourier transform; the first-diffraction-order band-limited intensity I(u) at the focal plane is measured by a CCD. To evaluate the Hamiltonian, a real distribution function g_c(u) is preset and the correlation F = ∫ I(u) g_c(u) du is computed. Its Fourier relation G(k) = ∫ g_c(u) sinc^2(Wu/λf) e^{iku} du ensures F = Σ_{j<h} G(j−h) σ_j^z σ_h^z, giving H = −F. Unlike prior target-intensity methods, g_c(u) may take both positive and negative values, enabling antiferromagnetic interactions (G < 0). For the number-partitioning problem, elements ζ_i ∈ (0,1] are labeled by spins σ_i ∈ {±1} to minimize H = (Σ_i ζ_i σ_i)^2 = Σ_{i,h} ζ_i ζ_h σ_i σ_h. The authors choose ξ_i = ζ_i and set uniform antiferromagnetic coupling G(k) = −1 so that σ_i' = ξ_i σ_i are encoded, and the Hamiltonian reduces to the total magnetization of gauge-transformed spins m' = Σ_i σ_i'. Experimentally, each spin is encoded by a 2×2 SLM macropixel of size W = 16 μm. A collimated 532 nm Gaussian beam is expanded (L2=50 mm, L3=500 mm), polarized along the SLM axis, and Fourier transformed by L1 (f = 100 mm). The CCD captures I(u). The origin of I(u) is calibrated by measuring with uniform SLM phase to mitigate optical aberrations. For a given G(k), g_c(u) is obtained numerically via inverse Fourier transform. A measurement-feedback loop implements a Markov chain Monte Carlo update: starting from uniform spins, tentative spin flips are encoded on the SLM, I(u) is measured, F (hence H) is computed via correlation, and updates are accepted if H decreases.

• Demonstrated antiferromagnetic Mattis model implementation in SPIM via correlation computing with gauge transformation, which cannot be achieved by the previous target-intensity approach due to the need for g_c(u) having both positive and negative values.
• Number-partitioning problem with N = 40,000 elements (200×200 array) experimentally tested. From four independent trials starting with uniform spins, the Hamiltonian H and gauge-transformed magnetization |m'| rapidly decrease initially, then stabilize; measurement noise can induce late fluctuations.
• Final |m'| achieved below 1.7 × 10^−3 within 100 iterations, indicating nearly three orders of magnitude reduction in magnetization during ground-state search.
• Scalability study across N = 1,600; 2,500; 6,400; 10,000; 16,900; 40,000 shows the averaged computing fidelity after 1,000 iterations remains within 6.9 × 10^−3, with error bars indicating standard error of the mean; fidelity remains stable versus N despite noise accumulation.
• Computational analysis shows optical computation dominates: for N = 40,000, more than 94% of total computation corresponds to the physical process of light, highlighting ultrafast, high-throughput, low-power advantages.
• Practical calibration of I(u) origin using uniform SLM phase addresses optical aberrations; g_c(u) numerically obtained for G(k) = −1; measured intensity maps used for correlation evaluation.

The work addresses the open challenge of implementing antiferromagnetic interactions in SPIM by introducing correlation-based optoelectronic evaluation of the Hamiltonian coupled with a gauge transformation that encodes both spins and couplings on a single phase-only SLM. This enables evaluation of Mattis-type antiferromagnetic models where the interaction kernel requires positive and negative contributions, overcoming the constraints of target-intensity approaches. Experimental results on large-scale number-partitioning demonstrate rapid descent in H and m' and robust scalability up to N = 40,000, validating that the SPIM with measurement-feedback effectively steers the system toward ground states despite optical aberrations and measurement uncertainties. The observed late-iteration fluctuations are explained by weak I(u) signals and CCD noise, suggesting improvements with higher dynamic-range, lower-noise detectors or adaptive input intensity control. The dominance of optical computation (>94% at N=40,000) underscores the method’s potential for accelerated, energy-efficient analog computation of antiferromagnetic Hamiltonians and combinatorial problems. The approach is compatible with various ground-state-search strategies (e.g., simulated annealing, adiabatic evolution) and benefits from ultrafast SLM/CCD hardware for further speed gains.

The authors propose and experimentally realize an antiferromagnetic spatial photonic Ising machine using optoelectronic correlation computing with a gauge transformation, enabling encoding of spins and couplings on a single phase-only SLM. They demonstrate ground-state-search acceleration for a Mattis-type antiferromagnetic model and the NP-hard number-partitioning problem with up to 40,000 spins, achieving nearly three orders of magnitude reduction in gauge-transformed magnetization and stable fidelity across system sizes. The method exhibits strong programmability, scalability, and optical-computation dominance. Future improvements include using ultrafast SLMs and CCDs operating at gigahertz rates to boost speed, more sensitive low-noise detectors to enhance accuracy, and applying the method within adiabatic evolution and simulated annealing frameworks for broader classes of optimization and statistical physics problems.

• Measurement noise and limited dynamic range of the CCD can lead to incorrect spin updates in late iterations when I(u) is weak, causing fluctuations in m'.
• Optical aberrations require calibration (setting SLM uniform phase and fitting maximal intensity) to align the origin between g_c(u) and I(u).
• Fidelity stability across increasing N is constrained by accumulating measurement uncertainty; background noise is a limiting factor.
• Implementation relies on numerical computation of the correlation integral and g_c(u); while optical processing dominates, digital components introduce overhead.
• The target-intensity approach cannot realize antiferromagnetic cases due to the necessity of negative g_c(u), mandating the correlation-computing scheme.
• Performance enhancements depend on availability of ultrafast, low-noise SLM/CCD hardware and potentially adaptive intensity control.