Accelerating ground state search of spatial photonic Ising machines with genetic-simulated annealing hybrid algorithm

Equilibrium Propagation offers a compelling alternative to traditional machine learning for training energy-based networks. Here we demonstrate a hybrid optical-digital implementation of EP using a Spatial Photonic Ising Machine (SPIM). The SPIM exploits the gauge transformation method to optically encode both continuous neuron states and rank-1 binary trainable patterns as phase modulations via a spatial light modulator, with inference realized using a finite difference scheme. The experimental system is evaluated on the Wine classification dataset. The potential of this approach, including the use of continuous couplings and structured coupling matrices, is evaluated numerically on the more complex MNIST dataset. Our work provides a concrete pathway toward energy-efficient physical implementations of Equilibrium Propagation.

Ising machines -- special-purpose hardware for heuristically solving Ising optimization problems -- based on probabilistic bits (p-bits) have been established as a promising alternative to heuristic optimization algorithms run on conventional computers. However, it has -- until now -- been thought that Ising spins that are connected in probabilistic Ising machines cannot be updated in parallel without ruining the machine's solving ability. This has been a major challenge for using probabilistic Ising machines as fast solvers for densely connected problems. Here, we circumvent this by introducing a modified Ising spin dynamics with an added inertia term, and verify in algorithm simulations, FPGA hardware emulation, and FPGA experiments that it enables fully parallel, synchronous updates while improving rather than degrading success probability. We evaluated on various types of abstract (Max-Cut and Sherrington-Kirkpatrick-model) and application-derived (MIMO, wireless detection) dense Ising benchmark instances. Performing fully parallel updates results in a speed advantage that grows faster than linearly with the number of spins, giving rise to large time-to-solution increases for practical problem sizes. For both Max-Cut and the SK-1 model at a problem size of 200, our approach achieved an average speedup of $\approx 35\times$, with the best single-instance speedup reaching $150\times$. As an example of the practical utility of our approach in an application where speed is critical, we further show by co-designing the algorithm dynamics with the hardware implementation -- co-optimizing for solver ability and silicon resource usage -- that probabilistic Ising machines based on our approach satisfy the stringent solution quality and latency/throughput requirements for real-time MIMO detection in modern 5G cellular wireless networks while using a practically reasonable silicon area.

We study the use of graph neural networks (GNNs) for finding approximate ground states of Ising models. Efficiently finding these ground states is of broad significance because many combinatorial optimization problems can be formulated as an Ising model with the appropriate choice of couplings and fields. Exactly solving these problems is hard but there are many good heuristic methods. A lineage of these heuristics build from mean-field approximations: one approach uses the leading eigenvector of an appropriately defined matrix, another is the min-sum algorithm, also known as the energetic cavity method. Without modification, GNNs perform worse than both of these methods. We consider small modifications to the GNN to incorporate these heuristics and find that this considerably improves performance. While the modified approach is competitive against other deep-learning approaches, we still find that simulated annealing is reliably at least as good as deep learning methods for the same computational cost.