# Quantum Annealing Formulation for Binary Neural Networks

arXiv: https://arxiv.org/abs/2107.02751 · HTML: https://mlsanity.com/paper/2107.02751/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 35 references that are in the map.
- First posted on arXiv 5 Jul 2021.

## Builds on (most cited first, 12 of 12)

- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks](https://mlsanity.com/paper/1603.05279/) (2016)
- [Binary Neural Networks: A Survey](https://mlsanity.com/paper/2004.03333/) (2020)
- [Efficient Learning for Deep Quantum Neural Networks](https://mlsanity.com/paper/1902.10445/) (2019)
- [XNOR-Net++: Improved Binary Neural Networks](https://mlsanity.com/paper/1909.13863/) (2019)
- [Quantum Variational Autoencoder](https://mlsanity.com/paper/1802.05779/) (2018)
- [Support vector machines on the D-Wave quantum annealer](https://mlsanity.com/paper/1906.06283/) (2019)
- [XNOR Neural Engine: a Hardware Accelerator IP for 21.6 fJ/op Binary Neural Network Inference](https://mlsanity.com/paper/1807.03010/) (2018)
- [A practical heuristic for finding graph minors](https://mlsanity.com/paper/1406.2741/) (2014)
- [Adiabatic Quantum Linear Regression](https://mlsanity.com/paper/2008.02355/) (2020)
- [A Path Towards Quantum Advantage in Training Deep Generative Models with Quantum Annealers](https://mlsanity.com/paper/1912.02119/) (2019)
- [Quantum-assisted associative adversarial network: Applying quantum annealing in deep learning](https://mlsanity.com/paper/1904.10573/) (2019)

## Built on by (most cited first, 15 of 15)

- [A Hybrid Quantum-Classical Algorithm for Robust Fitting](https://mlsanity.com/paper/2201.10110/) (2022)
- [Completely Quantum Neural Networks](https://mlsanity.com/paper/2202.11727/) (2022)
- [Quantum-enhanced Computer Vision: Going Beyond Classical Algorithms](https://mlsanity.com/paper/2510.07317/) (2025)
- [QuAnt: Quantum Annealing with Learnt Couplings](https://mlsanity.com/paper/2210.08114/) (2022)
- [Accelerating the training of single-layer binary neural networks using the HHL quantum algorithm](https://mlsanity.com/paper/2210.12707/) (2022)
- [Projected Stochastic Gradient Descent with Quantum Annealed Binary Gradients](https://mlsanity.com/paper/2310.15128/) (2023)
- [QUBO Decision Tree: Annealing Machine Extends Decision Tree Splitting](https://mlsanity.com/paper/2303.09772/) (2023)
- [An Advantage Using Feature Selection with a Quantum Annealer](https://mlsanity.com/paper/2211.09756/) (2022)
- [Quantum Neuron Selection: Finding High Performing Subnetworks With Quantum Algorithms](https://mlsanity.com/paper/2302.05984/) (2023)
- [Training Multilayer Perceptrons by Sampling with Quantum Annealers](https://mlsanity.com/paper/2303.12352/) (2023)
- [Outlier-Robust Multi-Model Fitting on Quantum Annealers](https://mlsanity.com/paper/2504.13836/) (2025)
- [αQBoost: An Iteratively Weighted Adiabatic Trained Classifier](https://mlsanity.com/paper/2210.07984/) (2022)
- [QUBO-inspired Molecular Fingerprint for Chemical Property Prediction](https://mlsanity.com/paper/2303.10179/) (2023)
- [Quadratic Unconstrained Binary Optimisation for Training and Regularisation of Binary Neural Networks](https://mlsanity.com/paper/2601.00449/) (2026)
- [Scalable Digital Compute-in-Memory Ising Machines for Robustness Verification of Binary Neural Networks](https://mlsanity.com/paper/2603.05677/) (2026)

## Abstract (arXiv)

Quantum annealing is a promising paradigm for building practical quantum computers. Compared to other approaches, quantum annealing technology has been scaled up to a larger number of qubits. On the other hand, deep learning has been profoundly successful in pushing the boundaries of AI. It is thus natural to investigate potentially game changing technologies such as quantum annealers to augment the capabilities of deep learning. In this work, we explore binary neural networks, which are lightweight yet powerful models typically intended for resource constrained devices. Departing from current training regimes for binary networks that smooth/approximate the activation functions to make the network differentiable, we devise a quadratic unconstrained binary optimization formulation for the training problem. While the problem is intractable, i.e., the cost to estimate the binary weights scales exponentially with network size, we show how the problem can be optimized directly on a quantum annealer, thereby opening up to the potential gains of quantum computing. We experimentally validated our formulation via simulation and testing on an actual quantum annealer (D-Wave Advantage), the latter to the extent allowable by the capacity of current technology.