# A Hybrid Quantum-Classical Approach based on the Hadamard Transform for the Convolutional Layer

arXiv: https://arxiv.org/abs/2305.17510 · HTML: https://mlsanity.com/paper/2305.17510/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 45 references that are in the map.
- First posted on arXiv 27 May 2023.

## Builds on (most cited first, 11 of 11)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Harmonic Convolutional Networks based on Discrete Cosine Transform](https://mlsanity.com/paper/2001.06570/) (2020)
- [Fast Walsh-Hadamard Transform and Smooth-Thresholding Based Binary Layers in Deep Neural Networks](https://mlsanity.com/paper/2104.07085/) (2021)
- [Block Walsh-Hadamard Transform Based Binary Layers in Deep Neural Networks](https://mlsanity.com/paper/2201.02711/) (2022)
- [Energy Efficient Hadamard Neural Networks](https://mlsanity.com/paper/1805.05421/) (2018)
- [Less is More: Accelerating Faster Neural Networks Straight from JPEG](https://mlsanity.com/paper/2104.00185/) (2021)
- [Discrete Cosine Transform Based Causal Convolutional Neural Network for Drift Compensation in Chemical Sensors](https://mlsanity.com/paper/2011.06681/) (2020)
- [A Simulation Study to Evaluate the Performance of the Cauchy Proximal Operator in Despeckling SAR Images of the Sea Surface](https://mlsanity.com/paper/2012.06657/) (2020)
- [Real-time Wireless ECG-derived Respiration Rate Estimation Using an Autoencoder with a DCT Layer](https://mlsanity.com/paper/2211.08491/) (2022)
- [HadaNets: Flexible Quantization Strategies for Neural Networks](https://mlsanity.com/paper/1905.10759/) (2019)
- [A hybrid classical-quantum algorithm for digital image processing](https://mlsanity.com/paper/2208.09714/) (2022)

## Built on by (most cited first, 12 of 12)

- [Domain Generalization with Fourier Transform and Soft Thresholding](https://mlsanity.com/paper/2309.09866/) (2023)
- [A Probabilistic Hadamard U-Net for MRI Bias Field Correction](https://mlsanity.com/paper/2403.05024/) (2024)
- [ADC/DAC-Free Analog Acceleration of Deep Neural Networks with Frequency Transformation](https://mlsanity.com/paper/2309.01771/) (2023)
- [A novel asymmetrical autoencoder with a sparsifying discrete cosine Stockwell transform layer for gearbox sensor data compression](https://mlsanity.com/paper/2310.02862/) (2023)
- [Discrete Cosine Transform Based Decorrelated Attention for Vision Transformers](https://mlsanity.com/paper/2405.13901/) (2024)
- [Qsco: A Quantum Scoring Module for Open-set Supervised Anomaly Detection](https://mlsanity.com/paper/2405.16368/) (2024)
- [HTMA-Net: Towards Multiplication-Avoiding Neural Networks via Hadamard Transform and In-Memory Computing](https://mlsanity.com/paper/2509.23103/) (2025)
- [The Blind Normalized Stein Variational Gradient Descent-Based Detection for Intelligent Random Access in Cellular IoT](https://mlsanity.com/paper/2403.18846/) (2024)
- [Enhancing Microgrid Performance Prediction with Attention-based Deep Learning Models](https://mlsanity.com/paper/2407.14984/) (2024)
- [WTHaar-Net: a Hybrid Quantum-Classical Approach](https://mlsanity.com/paper/2603.02497/) (2026)
- [Structure of the Circular-Dyadic Convolution Error](https://mlsanity.com/paper/2607.15293/) (2026)
- [Learned Queries and Keys Are All You Need: Replacing the Value Projection with Structured Transforms](https://mlsanity.com/paper/2609.36698/) (2026)

## Abstract (arXiv)

In this paper, we propose a novel Hadamard Transform (HT)-based neural network layer for hybrid quantum-classical computing. It implements the regular convolutional layers in the Hadamard transform domain. The idea is based on the HT convolution theorem which states that the dyadic convolution between two vectors is equivalent to the element-wise multiplication of their HT representation. Computing the HT is simply the application of a Hadamard gate to each qubit individually, so the HT computations of our proposed layer can be implemented on a quantum computer. Compared to the regular Conv2D layer, the proposed HT-perceptron layer is computationally more efficient. Compared to a CNN with the same number of trainable parameters and 99.26% test accuracy, our HT network reaches 99.31% test accuracy with 57.1% MACs reduced in the MNIST dataset; and in our ImageNet-1K experiments, our HT-based ResNet-50 exceeds the accuracy of the baseline ResNet-50 by 0.59% center-crop top-1 accuracy using 11.5% fewer parameters with 12.6% fewer MACs.