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 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 ≈35×, with the best single-instance speedup reaching 150×. 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.
The generalized Ising problem captures a broad spectrum of hard combinatorial problems, including MAX-CUT, Number Partitioning (NPP), and Maximum Independent Set. In this work, we consider the notion of one-flip local minima for this problem. We construct a polynomial relaxation and prove the landscape equivalence theorem: there exists a one-to-one correspondence between the local minima of the relaxation and the one-flip minima of the original Ising problem. This guarantee reduces the Ising problem to finding the local minima of a smooth function, allowing us to leverage gradient-based optimizers such as ADAM. We demonstrate that our method is scalable and it achieves strong performance across challenging benchmarks, including spin-glass models, MAX-CUT, and NPP.
Debraj Banerjee, Santanu Mahapatra, Kunal N. Chaudhury
While Ising machines serve as advanced physical solvers for the Ising model,enabling applications in combinatorial optimization and neural network training,their scalability for large-scale neural networks remains constrained by hardware connectivity limitations and suboptimal training methodologies. In this work,we leverage a Coherent Ising Machine (CIM) to train an energy-based neural network using Equilibrium Propagation, achieving performance comparable to existing software-based implementations. We further enhance the algorithm by integrating the Adam optimizer to solve for the ground state of a Hopfield energy network, significantly improving convergence speed and solution accuracy. Additionally, we demonstrate the scalability of our approach across deeper network architectures and convolutional operations. Our results highlight the potential of CIM dynamics as a scalable platform for training complex neural networks, offering a pathway toward energy-efficient implementations via analog circuits, optoelectronics, or integrated photonics. This work establishes a novel physical framework for next-generation AI hardware development.