# Noise-injected analog Ising machines enable ultrafast statistical sampling and machine learning

arXiv: https://arxiv.org/abs/2112.11534 · HTML: https://mlsanity.com/paper/2112.11534/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 55 references that are in the map.
- First posted on arXiv 21 Dec 2021.

## Builds on (most cited first, 3 of 3)

- [Energy and Policy Considerations for Deep Learning in NLP](https://mlsanity.com/paper/1906.02243/) (2019)
- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Quantum-inspired annealers as Boltzmann generators for machine learning and statistical physics](https://mlsanity.com/paper/1912.08480/) (2019)

## Built on by (most cited first, 11 of 11)

- [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)
- [Multi-Scale Simulation of Complex Systems: A Perspective of Integrating Knowledge and Data](https://mlsanity.com/paper/2306.10275/) (2023)
- [Low-rank combinatorial optimization and statistical learning by spatial photonic Ising machine](https://mlsanity.com/paper/2303.14993/) (2023)
- [Training Multi-layer Neural Networks on Ising Machine](https://mlsanity.com/paper/2311.03408/) (2023)
- [From Independent to Correlated Diffusion: Generalized Generative Modeling with Probabilistic Computers](https://mlsanity.com/paper/2603.27996/) (2026)
- [Mean-Field Assisted Deep Boltzmann Learning with Probabilistic Computers](https://mlsanity.com/paper/2401.01996/) (2024)
- [Photonic restricted Boltzmann machine for content generation tasks](https://mlsanity.com/paper/2508.20472/) (2025)
- [Training Energy-Based Models with Non-MCMC Samplers and Efficient Temperature Estimation](https://mlsanity.com/paper/2512.02323/) (2025)
- [Beyond Gradient Descent: Adam for Analog Ising Machines](https://mlsanity.com/paper/2606.03917/) (2026)
- [Optimizing Energy-based Neural Network Training with Coherent Ising Machine](https://mlsanity.com/paper/2606.09117/) (2026)

## Abstract (arXiv)

Ising machines are a promising non-von-Neumann computational concept for neural network training and combinatorial optimization. However, while various neural networks can be implemented with Ising machines, their inability to perform fast statistical sampling makes them inefficient for training neural networks compared to digital computers. Here, we introduce a universal concept to achieve ultrafast statistical sampling with analog Ising machines by injecting noise. With an opto-electronic Ising machine, we experimentally demonstrate that this can be used for accurate sampling of Boltzmann distributions and for unsupervised training of neural networks, with equal accuracy as software-based training. Through simulations, we find that Ising machines can perform statistical sampling orders-of-magnitudes faster than software-based methods. This enables the use of Ising machines beyond combinatorial optimization and makes them into efficient tools for machine learning and other applications.