Reducing hyperparameter sensitivity in measurement-feedback based Ising machines

As Moore's law reaches its limits, Ising machines offer a promising alternative computing approach for difficult optimization problems. However, many analog, time-continuous Ising machines rely on gradient-descent-like dynamics to find solutions, which can limit speed and robustness. We investigate whether momentum and Adam optimization can improve these systems. Since these optimizers are traditionally formulated in discrete time, we derive continuous-time versions suitable for analog, time-continuous Ising-machine dynamics. On Max-Cut benchmarks, we find that Adam-based dynamics substantially reduce time-to-target and improve solution quality compared with gradient-descent- and momentum-based dynamics. We further introduce a first-order continuous-time approximation of Adam that is intended as a simpler starting point for future physical implementations and while performing better than the full Adam formulation in a continuous-time setting. We also study a purely algorithmic discrete-time setting, where the performance gap is reduced on easier problem instances, while the Adam-based update rule performs best on harder weighted problem instances. These results identify continuous-time Adam dynamics as a powerful design principle for analog Ising machines.

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 (PIMs) cannot be updated in parallel without ruining the machine's solving ability. This has presented a major challenge to realizing the potential for probabilistic Ising machines to act as fast solvers for densely connected Ising problems. In this paper, we show that it is possible to circumvent this conventional wisdom. We introduce a modified form of Ising spin dynamics for PIMs, adding an inertia term, and verify in algorithm simulations, field-programmable gate array (FPGA) emulation, and in FPGA experiments that the modified dynamics enables fully parallel, synchronous updates and at the same time improves the achieved success probability. Our evaluations were performed with various types of abstract (Max-Cut and Sherrington-Kirkpatrick model) and application-derived (multiple-input and multiple-output, MIMO detection) dense Ising benchmark instances. Performing fully parallel updates results in a speed advantage that grows superlinearly with the number of spins, giving rise to large time-to-solution reductions for practical problem sizes. For both MC and the SK model at a problem size of 200, our approach achieved an average speedup of ~34x, with the best single-instance speedup reaching 150x. As an example of the practical utility of our approach in an application where speed is critical, we co-design the algorithm dynamics and hardware implementation for MIMO detection, achieving improved detection accuracy relative to the standard linear detector and higher throughput than the conventional sequential-update PIM.

Ising machines have emerged as promising hardware accelerators for intractable optimization and sampling problems, yet their practical impact increasingly hinges on the co-design of algorithms and hardware, where algorithmic demands shape new architectures and new hardware capabilities inspire entirely new algorithms. In this Review, we survey probabilistic algorithms designed for portability across diverse Ising platforms, advocating a top-down perspective that prioritizes principled methods with provable guarantees. We cover foundational methods such as simulated annealing and parallel tempering, including two-dimensional extensions that natively encode hard constraints, and examine approaches that expand the scale of solvable problems from cluster mean-field methods to variational samplers. We highlight the Probabilistic Approximate Optimization Algorithm (PAOA), a classical analog of QAOA that emerged directly from probabilistic hardware development, and explore how generative AI and Ising machines might reinforce each other: learned models propose global moves to accelerate optimization, while probabilistic techniques improve inference in large language models. Much as quantum computing has seen algorithms co-evolve with hardware, probabilistic and Ising computing stand at a similar inflection point. We outline a co-design framework for accelerating the capabilities and adoption of next-generation Ising machines.