# Projected Stochastic Gradient Descent with Quantum Annealed Binary Gradients

arXiv: https://arxiv.org/abs/2310.15128 · HTML: https://mlsanity.com/paper/2310.15128/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 75 references that are in the map.
- First posted on arXiv 23 Oct 2023.

## Builds on (most cited first, 30 of 48)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks](https://mlsanity.com/paper/1603.05279/) (2016)
- [A Survey of Quantization Methods for Efficient Neural Network Inference](https://mlsanity.com/paper/2103.13630/) (2021)
- [Large Batch Optimization for Deep Learning: Training BERT in 76 minutes](https://mlsanity.com/paper/1904.00962/) (2019)
- [BinaryConnect: Training Deep Neural Networks with binary weights during propagations](https://mlsanity.com/paper/1511.00363/) (2015)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [DoReFa-Net: Training Low Bitwidth Convolutional Neural Networks with Low Bitwidth Gradients](https://mlsanity.com/paper/1606.06160/) (2016)
- [A White Paper on Neural Network Quantization](https://mlsanity.com/paper/2106.08295/) (2021)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [PACT: Parameterized Clipping Activation for Quantized Neural Networks](https://mlsanity.com/paper/1805.06085/) (2018)
- [Deep Learning on Graphs: A Survey](https://mlsanity.com/paper/1812.04202/) (2018)
- [LQ-Nets: Learned Quantization for Highly Accurate and Compact Deep Neural Networks](https://mlsanity.com/paper/1807.10029/) (2018)
- [Bi-Real Net: Enhancing the Performance of 1-bit CNNs With Improved Representational Capability and Advanced Training Algorithm](https://mlsanity.com/paper/1808.00278/) (2018)
- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Binary Neural Networks: A Survey](https://mlsanity.com/paper/2004.03333/) (2020)
- [ReActNet: Towards Precise Binary Neural Network with Generalized Activation Functions](https://mlsanity.com/paper/2003.03488/) (2020)
- [Efficient Learning for Deep Quantum Neural Networks](https://mlsanity.com/paper/1902.10445/) (2019)
- [Cooperative Game Theory Approaches for Network Partitioning](https://mlsanity.com/paper/1707.03587/) (2017)
- [Deep Learning with Low Precision by Half-wave Gaussian Quantization](https://mlsanity.com/paper/1702.00953/) (2017)
- [XNOR-Net++: Improved Binary Neural Networks](https://mlsanity.com/paper/1909.13863/) (2019)
- [TensorFlow Quantum: A Software Framework for Quantum Machine Learning](https://mlsanity.com/paper/2003.02989/) (2020)
- [ProxQuant: Quantized Neural Networks via Proximal Operators](https://mlsanity.com/paper/1810.00861/) (2018)
- [Loss-aware Binarization of Deep Networks](https://mlsanity.com/paper/1611.01600/) (2016)
- [A comprehensive review of Binary Neural Network](https://mlsanity.com/paper/2110.06804/) (2021)
- [Application of Quantum Annealing to Training of Deep Neural Networks](https://mlsanity.com/paper/1510.06356/) (2015)
- [Binary Graph Neural Networks](https://mlsanity.com/paper/2012.15823/) (2020)
- [QUBO Formulations for Training Machine Learning Models](https://mlsanity.com/paper/2008.02369/) (2020)

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

- [Quantum-enhanced Computer Vision: Going Beyond Classical Algorithms](https://mlsanity.com/paper/2510.07317/) (2025)
- [Outlier-Robust Multi-Model Fitting on Quantum Annealers](https://mlsanity.com/paper/2504.13836/) (2025)
- [Layered Quantum Architecture Search for 3D Point Cloud Classification](https://mlsanity.com/paper/2603.20024/) (2026)

## Abstract (arXiv)

We present, QP-SBGD, a novel layer-wise stochastic optimiser tailored towards training neural networks with binary weights, known as binary neural networks (BNNs), on quantum hardware. BNNs reduce the computational requirements and energy consumption of deep learning models with minimal loss in accuracy. However, training them in practice remains to be an open challenge. Most known BNN-optimisers either rely on projected updates or binarise weights post-training. Instead, QP-SBGD approximately maps the gradient onto binary variables, by solving a quadratic constrained binary optimisation. Under practically reasonable assumptions, we show that this update rule converges with a rate of O(1 / √T). Moreover, we show how the NP-hard projection can be effectively executed on an adiabatic quantum annealer, harnessing recent advancements in quantum computation. We also introduce a projected version of this update rule and prove that if a fixed point exists in the binary variable space, the modified updates will converge to it. Last but not least, our algorithm is implemented layer-wise, making it suitable to train larger networks on resource-limited quantum hardware. Through extensive evaluations, we show that QP-SBGD outperforms or is on par with competitive and well-established baselines such as BinaryConnect, signSGD and ProxQuant when optimising the Rosenbrock function, training BNNs as well as binary graph neural networks.