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Optical Ising machines with two-body interactions have shown potential in solving combinatorial optimization problems which are extremely hard to solve with digital computers. Yet, some physical systems cannot be properly described by only two-body interactions. Here, we propose and demonstrate a nonlinear optics approach to emulate Ising machines containing many spins (up to a million in the absence of optical imperfections) and with tailored all-to-all two and four-body interactions. Ourdoi:10.1038/s42005-020-0376-5 fatcat:r7vo7ln7sjg6tpqv2zow3pcgse