Spatial-photonic Boltzmann machines: low-rank combinatorial optimization and statistical learning by spatial light modulation

The spatial-photonic Ising machine (SPIM) [D. Pierangeli et al., Phys. Rev. Lett. 122, 213902 (2019)] is a promising optical architecture utilizing spatial light modulation for solving large-scale combinatorial optimization problems efficiently. However, the SPIM can accommodate Ising problems with only rank-one interaction matrices, which limits its applicability to various real-world problems. In this Letter, we propose a new computing model for the SPIM that can accommodate any Ising problem without changing its optical implementation. The proposed model is particularly efficient for Ising problems with low-rank interaction matrices, such as knapsack problems. Moreover, the model acquires learning ability and can thus be termed a spatial-photonic Boltzmann machine (SPBM). We demonstrate that learning, classification, and sampling of the MNIST handwritten digit images are achieved efficiently using SPBMs with low-rank interactions. Thus, the proposed SPBM model exhibits higher practical applicability to various problems of combinatorial optimization and statistical learning, without losing the scalability inherent in the SPIM architecture.