Probabilistic Algorithms for Ising Machines from Optimization to Generative AI

Corentin Delacour; Xiuqi Zhang; Abdelrahman S. Abdelrahman; Saleh Bunaiyan; Kyle Lee; Shuvro Chowdhury; Kerem Y. Camsari

cs.LGOct 2, 2026

Probabilistic Algorithms for Ising Machines from Optimization to Generative AI

Authors: Corentin Delacour, Xiuqi Zhang, Abdelrahman S. Abdelrahman, Saleh Bunaiyan, Kyle Lee, Shuvro Chowdhury, Kerem Y. Camsari

Organizations: Department of Electrical and Computer Engineering, University of California, Santa Barbara, Santa Barbara, CA 93106, USA · Electrical Engineering Department, King Fahd University of Petroleum & Minerals (KFUPM), Dhahran 31261, Saudi Arabia

Abstract

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.

Figures & tables

Figure 1: Ising machine workflow and probabilistic algorithm-hardware co-design. (a) Top: an optimization or sampling task is encoded as an Ising problem with energy $E$ and Boltzmann distribution $S\sim\exp(-E/T)$ with $T$ the temperature parameter. Middle: the algorithm and Ising machine co-design loop, where algorithmic demands set hardware parameters and hardware constraints shape algorithm choice. Bottom: the Ising machine explores the energy landscape over time and returns samples $S$ . (b) Three families of probabilistic algorithms surveyed in this Review, paired with their hardware requirements. Simulated annealing requires time-modulation of temperature via a scheduler. Parallel tempering fixes distinct temperatures across replicas in space and requires energy calculation and replica synchronization for state swaps. Variational approaches couple an Ising machine with a classical optimizer that iteratively updates parameters based on sampled states. The Probabilistic Approximate Optimization Algorithm (PAOA) showcases this by learning temperatures across layers.

Figure 2: Cluster mean-field theory for hardware capacity scaling. (a) An original interaction graph whose size exceeds the capacity of a single Ising-machine chip. (b) Balanced partitioning of the graph into clusters sized to fit within hardware limits. (c) Serial CMFT workflow: the active cluster is updated on the Ising machine while remaining clusters stay frozen. Boundary interactions are approximated by mean-field values computed on a classical processor. (d) CPU results for $L=6$ (216 spins): residual energy versus Monte Carlo sweeps for different cluster counts (100 instances, 20 runs). Fewer clusters yield steeper power-law decay, where $\kappa$ denotes the log–log slope (exact graph: $\kappa=0.43$ ; two clusters: $\kappa=0.40$ ; three clusters: $\kappa=0.29$ ), while naive mean-field (each spin as its own cluster) is nearly flat ( $\kappa=0.02$ ). (e) FPGA results for $L=22$ (10,648 spins): two-cluster CMFT enables problem sizes beyond single-FPGA capacity while reaching residual energies comparable to the unpartitioned CPU baseline at shorter wall-clock times (10 instances, 10 runs).

Figure 3: Overview of PAOA. A classical processor (left) evaluates a cost function using independent samples returned by an Ising machine or probabilistic sampler (right). The resulting objective is used to update variational parameters, such as effective couplings, fields, or annealing schedules, that define the stochastic dynamics for the next layer. Repeating this sample-and-update loop produces a learned non-equilibrium trajectory. In the illustrated ansatz, an $M\times M$ weight matrix and $M\times 1$ bias vector parameterize the sampling distribution over $N$ problem spins, where $M\geq N$ .

Figure 4: Synergies between generative AI and Ising machines. Left: Equilibrium samples collected from MCMC chains after burn-in serve as training data for a generative model. In this setup, a decoder-only Transformer (IsingFormer) is trained to produce equilibrium-like samples in fewer steps, effectively bypassing mixing times. Right: A Generator-Verifier scheme employing Transformer-Augmented Parallel Tempering (TAPT). Conditioned on the inverse temperature $\beta$ , the IsingFormer acts as a generator to accelerate PT exploration via global moves along the temperature ladder. PT functions as a verifier, accepting or rejecting moves based on a Metropolis criterion to enforce detailed balance. The cold replicas remain computationally intractable for direct generation.

Stochastic Optimization

Stochastic optimization studies how to find good decisions or model parameters when objectives, data, or feedback are random or noisy.

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Jun 17, 2026

A Tool for the Synthesis of Adaptive Probabilistic Processors Based on the Ising Model

This work presents a tool for the synthesis and simulation of probabilistic architectures for solving combinatorial optimization problems by mapping them to the Ising model. The proposed approach automatically constructs the Ising Hamiltonian and determines the number of probabilistic elements (p-bits) based on problem characteristics such as size and topology. Furthermore, the tool introduces an adaptive strategy for selecting the most suitable update algorithm among Gibbs Sampling, Simulated Annealing (SA), Simulated Quantum Annealing (SQA), and cluster-based methods. Experimental results using benchmark problems demonstrate improved convergence behavior and flexibility compared to fixed approaches. The proposed framework enables systematic evaluation of probabilistic computing strategies and supports the development of future hardware implementations based on MTJs and p-bits.

Jonathan Juracy Carneiro da Silva, Leonardo R. Gobatto, Jose Rodrigo Azambuja

Federal University of Rio Grande do Sul (UFRGS) – Institute of Informatics – PGMicro, Porto Alegre, Brazil · Center for Embedded Devices and Research in Digital Agriculture (CEDRA), S˜ao Leopoldo, Brazil

Algorithm Selection Metaheuristic Optimization Combinatorial Optimization

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Apr 18, 2026

A fully parallel densely connected probabilistic Ising machine with inertia for real-time applications

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.

Ruomin Zhu, Abhishek Kumar Singh, Jérémie Laydevant +5

School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA · Department of Computer Science, Princeton University, Princeton, NJ, 08540, USA · USRA Research Institute for Advanced Computer Science, Moffett Field, CA, 94035, USA +1

Ising Optimization

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Jun 2, 2026

Beyond Gradient Descent: Adam for Analog 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.

Stijn Van Vooren, Guy Van der Sande, Guy Verschaffelt

Applied Physics research group, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium∗

Combinatorial Optimization Adaptive Gradient Methods Max-Cut Analog Computing

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Abstract

Stochastic Optimization

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