Incorporating rank-free coupling and external field via an amplitude-only modulated spatial photonic Ising machine

Ising machines have emerged as effective solvers for combinatorial optimization problems, such as NP-hard problems, machine learning, and financial modeling. Recent spatial photonic Ising machines (SPIMs) excel in multi-node optimization and spin glass simulations, leveraging their large-scale and fully connected characteristics. However, existing laser diffraction-based SPIMs usually sacrifice time efficiency or spin count to encode high-rank spin-spin coupling and external fields, limiting their scalability for real-world applications. Here, we demonstrate an amplitude-only modulated rank-free spatial photonic Ising machine (AR-SPIM) with 200 iterations per second. By re-formulating an arbitrary Ising Hamiltonian as the sum of Hadamard products, followed by loading the corresponding matrices/vectors onto an aligned amplitude spatial light modulator and digital micro-mirrors device, we directly map a 797-spin Ising model with external fields (nearly 9-bit precision, -255 to 255) into an incoherent light field, eliminating the need for repeated and auxiliary operations. Serving as encoding accuracy metrics, the linear coefficient of determination and Pearson correlation coefficient between measured light intensities and Ising Hamiltonians exceed 0.9800, with values exceed 0.9997 globally. The AR-SPIM achieves less than 0.3% error rate for ground-state search of biased Max-cut problems with arbitrary ranks and weights, enables complex phase transition observations, and facilitates scalable spin counts for sparse Ising problems via removing zero-valued Hadamard product terms. This reconfigurable AR-SPIM can be further developed to support large-scale machine-learning training and deployed for practical applications in discrete optimization and quantum many-body simulations.