Programmable k-local Ising Machines and all‑optical Kolmogorov-Arnold Networks on Photonic Platforms

Photonic computing offers energy-efficient acceleration for optimization and learning, yet combinatorial search and function approximation have typically required different devices and control stacks. We present a unified approach that supports both k-local Ising optimization and optical Kolmogorov–Arnold network (KAN) learning on a single photonic platform. The core is an SLM-centric primitive capable of implementing all-optical k-local Ising interactions and fully optical KAN layers within the same architecture. Our framework maps k-body products to per-window structural nonlinearities, analyzes calibration and training through general Jacobian models, and outlines practical implementation routes on several photonic platforms. The key idea is to exploit the structural nonlinearity of a nominally linear scatterer by adding a second relay pass through the same SLM: a folded 4f relay re-images the Fourier plane so each clique or channel occupies its own window with an independent second-pass phase patch. Optical propagation remains linear, but each window’s measured intensity becomes a programmable polynomial of the clique sum or projection amplitude. This mechanism yields intrinsic k-local couplings without nonlinear media, and simultaneously provides the many independent nonlinearities required for KAN layers, all trainable via in-situ physical gradients using only forward and adjoint frames. We describe implementations on spatial photonic Ising machines, VCSEL arrays, and Microsoft’s analog optical hardware, requiring only one added lens and fold (or an on-chip 4f loop). Our analysis characterizes calibration behavior and noise robustness for broad classes of per-window Jacobians and serves as a blueprint for future experiments.