# Massively Parallel Probabilistic Computing with Sparse Ising Machines

arXiv: https://arxiv.org/abs/2110.02481 · HTML: https://mlsanity.com/paper/2110.02481/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Oct 2021.

## Built on by (most cited first, 25 of 25)

- [Training an Ising Machine with Equilibrium Propagation](https://mlsanity.com/paper/2305.18321/) (2023)
- [Training Deep Boltzmann Networks with Sparse Ising Machines](https://mlsanity.com/paper/2303.10728/) (2023)
- [Thermodynamic Computing System for AI Applications](https://mlsanity.com/paper/2312.04836/) (2023)
- [Thermodynamic AI and the fluctuation frontier](https://mlsanity.com/paper/2302.06584/) (2023)
- [CMOS + stochastic nanomagnets: heterogeneous computers for probabilistic inference and learning](https://mlsanity.com/paper/2304.05949/) (2023)
- [IsingFormer: Augmenting Parallel Tempering With Learned Proposals](https://mlsanity.com/paper/2509.23043/) (2025)
- [Thermodynamic Bayesian Inference](https://mlsanity.com/paper/2410.01793/) (2024)
- [Enhancing In-vehicle Multiple Object Tracking Systems with Embeddable Ising Machines](https://mlsanity.com/paper/2410.14093/) (2024)
- [A Blueprint for Equilibrium-Based Differentiable Continuous-Variable Thermodynamic Computing](https://mlsanity.com/paper/2607.16183/) (2026)
- [Machine Learning Quantum Systems with Magnetic p-bits](https://mlsanity.com/paper/2310.06679/) (2023)
- [Self-Adaptive Ising Machines for Constrained Optimization](https://mlsanity.com/paper/2501.04971/) (2025)
- [Two-dimensional Parallel Tempering for Constrained Optimization](https://mlsanity.com/paper/2506.14781/) (2025)
- [Generalized Probabilistic Approximate Optimization Algorithm](https://mlsanity.com/paper/2507.07420/) (2025)
- [Probabilistic Computers for Neural Quantum States](https://mlsanity.com/paper/2512.24558/) (2025)
- [From Independent to Correlated Diffusion: Generalized Generative Modeling with Probabilistic Computers](https://mlsanity.com/paper/2603.27996/) (2026)
- [Improved Sparse Ising Optimization](https://mlsanity.com/paper/2311.09275/) (2023)
- [Mean-Field Assisted Deep Boltzmann Learning with Probabilistic Computers](https://mlsanity.com/paper/2401.01996/) (2024)
- [Machine Learning-assisted High-speed Combinatorial Optimization with Ising Machines for Dynamically Changing Problems](https://mlsanity.com/paper/2503.23966/) (2025)
- [Efficient Optimization Accelerator Framework for Multistate Ising Problems](https://mlsanity.com/paper/2505.20250/) (2025)
- [Energy-Efficient Supervised Learning with a Binary Stochastic Forward-Forward Algorithm](https://mlsanity.com/paper/2507.06461/) (2025)
- [Non-Negative Matrix Factorization Using Non-Von Neumann Computers](https://mlsanity.com/paper/2512.00675/) (2025)
- [GPU-accelerated simulated annealing based on p-bits with real-world device-variability modeling](https://mlsanity.com/paper/2601.14476/) (2026)
- [Enhanced Convergence in p-bit Based Simulated Annealing with Partial Deactivation for Large-Scale Combinatorial Optimization Problems](https://mlsanity.com/paper/2601.15561/) (2026)
- [CN101 - A Digital Thermodynamic Computer for Generative AI](https://mlsanity.com/paper/2608.00754/) (2026)
- [Probabilistic Algorithms for Ising Machines from Optimization to Generative AI](https://mlsanity.com/paper/2610.03972/) (2026)

## Abstract (arXiv)

Inspired by the developments in quantum computing, building domain-specific classical hardware to solve computationally hard problems has received increasing attention. Here, by introducing systematic sparsification techniques, we demonstrate a massively parallel architecture: the sparse Ising Machine (sIM). Exploiting sparsity, sIM achieves ideal parallelism: its key figure of merit - flips per second - scales linearly with the number of probabilistic bits (p-bit) in the system. This makes sIM up to 6 orders of magnitude faster than a CPU implementing standard Gibbs sampling. Compared to optimized implementations in TPUs and GPUs, sIM delivers 5-18x speedup in sampling. In benchmark problems such as integer factorization, sIM can reliably factor semiprimes up to 32-bits, far larger than previous attempts from D-Wave and other probabilistic solvers. Strikingly, sIM beats competition-winning SAT solvers (by 4-700x in runtime to reach 95% accuracy) in solving 3SAT problems. Even when sampling is made inexact using faster clocks, sIM can find the correct ground state with further speedup. The problem encoding and sparsification techniques we introduce can be applied to other Ising Machines (classical and quantum) and the architecture we present can be used for scaling the demonstrated 5,000-10,000 p-bits to 1,000,000 or more through analog CMOS or nanodevices.