# Accelerating the training of single-layer binary neural networks using the HHL quantum algorithm

arXiv: https://arxiv.org/abs/2210.12707 · HTML: https://mlsanity.com/paper/2210.12707/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 27 references that are in the map.
- First posted on arXiv 23 Oct 2022.

## Builds on (most cited first, 7 of 7)

- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [QUBO Formulations for Training Machine Learning Models](https://mlsanity.com/paper/2008.02369/) (2020)
- [Quantum Annealing Formulation for Binary Neural Networks](https://mlsanity.com/paper/2107.02751/) (2021)
- [Bayesian Deep Learning on a Quantum Computer](https://mlsanity.com/paper/1806.11463/) (2018)
- [Defence against adversarial attacks using classical and quantum-enhanced Boltzmann machines](https://mlsanity.com/paper/2012.11619/) (2020)
- [SWAP Test for an Arbitrary Number of Quantum States](https://mlsanity.com/paper/2110.13261/) (2021)

## Built on by (most cited first, 3 of 3)

- [Quantum HyperNetworks: Training Binary Neural Networks in Quantum Superposition](https://mlsanity.com/paper/2301.08292/) (2023)
- [Solving the encoding bottleneck: of the HHL algorithm, by the HHL algorithm](https://mlsanity.com/paper/2502.13534/) (2025)
- [Depth-Based Matrix Classification for the HHL Quantum Algorithm](https://mlsanity.com/paper/2505.22454/) (2025)

## Abstract (arXiv)

Binary Neural Networks are a promising technique for implementing efficient deep models with reduced storage and computational requirements. The training of these is however, still a compute-intensive problem that grows drastically with the layer size and data input. At the core of this calculation is the linear regression problem. The Harrow-Hassidim-Lloyd (HHL) quantum algorithm has gained relevance thanks to its promise of providing a quantum state containing the solution of a linear system of equations. The solution is encoded in superposition at the output of a quantum circuit. Although this seems to provide the answer to the linear regression problem for the training neural networks, it also comes with multiple, difficult-to-avoid hurdles. This paper shows, however, that useful information can be extracted from the quantum-mechanical implementation of HHL, and used to reduce the complexity of finding the solution on the classical side.