mlsanityData to 14 Oct 2026

Training an Ising Machine with Equilibrium Propagation

arXiv 2305.18321 ↗First posted 22 May 2023cs.NE cs.LG quant-phcited by 29

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Abstract (arXiv)

Ising machines, which are hardware implementations of the Ising model of coupled spins, have been influential in the development of unsupervised learning algorithms at the origins of Artificial Intelligence (AI). However, their application to AI has been limited due to the complexities in matching supervised training methods with Ising machine physics, even though these methods are essential for achieving high accuracy. In this study, we demonstrate a novel approach to train Ising machines in a supervised way through the Equilibrium Propagation algorithm, achieving comparable results to software-based implementations. We employ the quantum annealing procedure of the D-Wave Ising machine to train a fully-connected neural network on the MNIST dataset. Furthermore, we demonstrate that the machine's connectivity supports convolution operations, enabling the training of a compact convolutional network with minimal spins per neuron. Our findings establish Ising machines as a promising trainable hardware platform for AI, with the potential to enhance machine learning applications.

Key facts. Cited by 29 papers in the map (data through 14 Oct 2026). Builds on 26 of its 82 references that are in the map. First posted on arXiv 22 May 2023.

Builds on

26 of 82 references.

References · 26

  1. 2016XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networkscited by 849
  2. 2019Surrogate Gradient Learning in Spiking Neural Networkscited by 626
  3. 2016Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagationcited by 259
  4. 2016Direct Feedback Alignment Provides Learning in Deep Neural Networkscited by 240
  5. 2013Ising formulations of many NP problemscited by 207
  6. 2021Next Generation Reservoir Computingcited by 160
  7. 2017Towards Accurate Binary Convolutional Neural Networkcited by 149
  8. 2021Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systemscited by 128
  9. 2020Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Biascited by 64
  10. 2020Training End-to-End Analog Neural Networks with Equilibrium Propagationcited by 63
  11. 2022Holomorphic Equilibrium Propagation Computes Exact Gradients Through Finite Size Oscillationscited by 42
  12. 2023Backpropagation-free Training of Deep Physical Neural Networkscited by 42
14 more
  1. 2015Application of Quantum Annealing to Training of Deep Neural Networkscited by 37
  2. 2022Towards Scaling Difference Target Propagation by Learning Backprop Targetscited by 34
  3. 2019Updates of Equilibrium Prop Match Gradients of Backprop Through Time in an RNN with Static Inputcited by 33
  4. 2020EqSpike: Spike-driven Equilibrium Propagation for Neuromorphic Implementationscited by 31
  5. 2021Self-learning Machines based on Hamiltonian Echo Backpropagationcited by 28
  6. 2016Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Modelscited by 26
  7. 2021Massively Parallel Probabilistic Computing with Sparse Ising Machinescited by 25
  8. 2023Training Deep Boltzmann Networks with Sparse Ising Machinescited by 24
  9. 2020Training and Classification using a Restricted Boltzmann Machine on the D-Wave 2000Qcited by 18
  10. 2019Image classification using quantum inference on the D-Wave 2Xcited by 15
  11. 2023Thermodynamic AI and the fluctuation frontiercited by 15
  12. 2021Training Dynamical Binary Neural Networks with Equilibrium Propagationcited by 13
  13. 2021Noise-injected analog Ising machines enable ultrafast statistical sampling and machine learningcited by 11
  14. 2020A Hybrid Quantum enabled RBM Advantage: Convolutional Autoencoders For Quantum Image Compression and Generative Learningcited by 7

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 5; 2024: 7; 2025: 9; 2026: 8.

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  1. 2024Training of Physical Neural Networkscited by 49
  2. 2023Energy-based learning algorithms for analog computing: a comparative studycited by 37
  3. 2023Machine Learning Without a Processor: Emergent Learning in a Nonlinear Electronic Metamaterialcited by 20
  4. 2024Self-Contrastive Forward-Forward Algorithmcited by 17
  5. 2024Quantum Equilibrium Propagation: Gradient-Descent Training of Quantum Systemscited by 10
  6. 2024Quantum Equilibrium Propagation for efficient training of quantum systems based on Onsager reciprocitycited by 9
  7. 2023A universal approximation theorem for nonlinear resistive networkscited by 6
  8. 2024Training all-mechanical neural networks for task learning through in situ backpropagationcited by 5
  9. 2025Local Learning Rules for Out-of-Equilibrium Physical Generative Modelscited by 4
  10. 2023Training Multi-layer Neural Networks on Ising Machinecited by 3
  11. 2025Q-Detection: A Quantum-Classical Hybrid Poisoning Attack Detection Methodcited by 3
  12. 2025Learning at the Speed of Physics: Equilibrium Propagation on Oscillator Ising Machinescited by 3
17 more
  1. 2025Near-Equilibrium Propagation training in nonlinear wave systemscited by 3
  2. 2026Equilibrium Propagation for Non-Conservative Systemscited by 3
  3. 2026Training a Predictive Coding Network on ImageNet using Equilibrium Propagationcited by 3
  4. 2023Efficient and Flexible Neural Network Training through Layer-wise Feedback Propagationcited by 2
  5. 2025Convergence of energy-based learning in linear resistive networkscited by 2
  6. 2026A Blueprint for Equilibrium-Based Differentiable Continuous-Variable Thermodynamic Computingcited by 2
  7. 2024Unsupervised End-to-End Training with a Self-Defined Targetcited by 1
  8. 2025Probabilistic Quantum SVM Training on Ising Machinecited by 1
  9. 2025Correspondence Between Ising Machines and Neural Networkscited by 1
  10. 2026Local learning for stable backpropagation-free neural network training towards physical learningcited by 1
  11. 2024Estimation of Energy-dissipation Lower-bounds for Neuromorphic Learning-in-memorycited by 0
  12. 2025Harnessing uncertainty when learning through Equilibrium Propagation in neural networkscited by 0
  13. 2025Analog Physical Systems Can Exhibit Double Descentcited by 0
  14. 2026Equilibrium Propagation and Hamiltonian Inference in the Diffusive Fitzhugh-Nagumo Modelcited by 0
  15. 2026Hybridizing Equilibrium Propagation with Ising Machines for Efficient Energy-Based Learningcited by 0
  16. 2026Optimizing Energy-based Neural Network Training with Coherent Ising Machinecited by 0
  17. 2026Optical Implementation of Equilibrium Propagation Using Spatial Photonic Ising Machinescited by 0