Local-Minima-Preserving Continuous Relaxation of Ising Problems

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.

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.

We introduce a physics-inspired continuous relaxation framework that yields substantially improved solutions for NP-hard combinatorial optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), binary sparse coding, and planted-solution Ising models. By parameterizing discrete binary variables as continuous wave-like states on the complex unit circle, we inherently smooth highly non-convex energy landscapes. We show that representing binary variables as complex phases reveals an implicit regularization mechanism that promotes convergence toward discrete states. Extracting this mechanism yields significant improvements even within standard real-valued optimization frameworks, using this regularizer explicitly. Empirically, this regularization yields vastly higher ground-state convergence rates than standard real-valued alternatives. Our models achieved zero error in large-scale 160x160 QUBO tasks under severe noise (sigma=0.25), and outperformed traditional algorithms (OMP and LASSO) in underdefined sparse coding with perfect recovery at sigma=0.15. The solver's robustness was further validated by recovering exact ground-state configurations in 8 out of 11 rigorously engineered planted-solution benchmarks.