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

We unify k-local Ising optimization and optical KAN function learning on a single photonic platform, establishing a critical convergence point in optical computing that enables interleaved discrete-continuous workflows. We introduce a single spacial light modulator (SLM)-centric primitive that realizes, in one stroke, all-optical k-local Ising interactions and fully optical Kolmogorov-Arnold network (KAN) layers. The central idea is to convert structural nonlinearity of a nominally linear photonic scatterer into a per-window computational resource by adding one relay pass through the same spatial light modulator. A folded 4f relay reimages the first Fourier plane onto the SLM so that each chosen spin clique or ridge channel occupies a disjoint window with its own second-pass phase patch. Propagation remains linear in the optical field, yet the measured intensity in each window becomes a freely programmable polynomial of the clique sum or projection amplitude. This yields native, per-clique k-local couplings without nonlinear media and, in parallel, the many independent univariate nonlinearities required by KAN layers, all with in-situ physical gradients for training using two-frame (forward and adjoint) physical gradients. We outline implementation on spatial photonic Ising machines, injection-locked VCSEL arrays, and the Microsoft analog optical computers. In all cases the hardware change is one extra lens and a fold (or an on-chip 4f loop), enabling a minimal overhead, massively parallel route to high-order optical Ising optimization and trainable, all-optical KAN processing.