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WTHaar-Net: a Hybrid Quantum-Classical Approach

Vittorio Palladino, Tsai Idden, Ahmet Enis Cetin

TL;DR

This work introduces WTHaar-Net, a convolutional neural network that replaces the Hadamard Transform used in prior hybrid architectures with the Haar Wavelet Transform (HWT), and shows that the HWT admits a quantum realization using structured Hadamard gates, enabling its decomposition into unitary operations suitable for quantum circuits.

Abstract

Convolutional neural networks rely on linear filtering operations that can be reformulated efficiently in suitable transform domains. At the same time, advances in quantum computing have shown that certain structured linear transforms can be implemented with shallow quantum circuits, opening the door to hybrid quantum-classical approaches for enhancing deep learning models. In this work, we introduce WTHaar-Net, a convolutional neural network that replaces the Hadamard Transform used in prior hybrid architectures with the Haar Wavelet Transform (HWT). Unlike the Hadamard Transform, the Haar transform provides spatially localized, multi-resolution representations that align more closely with the inductive biases of vision tasks. We show that the HWT admits a quantum realization using structured Hadamard gates, enabling its decomposition into unitary operations suitable for quantum circuits. Experiments on CIFAR-10 and Tiny-ImageNet demonstrate that WTHaar-Net achieves substantial parameter reduction while maintaining competitive accuracy. On Tiny-ImageNet, our approach outperforms both ResNet and Hadamard-based baselines. We validate the quantum implementation on IBM Quantum cloud hardware, demonstrating compatibility with near-term quantum devices.

WTHaar-Net: a Hybrid Quantum-Classical Approach

TL;DR

This work introduces WTHaar-Net, a convolutional neural network that replaces the Hadamard Transform used in prior hybrid architectures with the Haar Wavelet Transform (HWT), and shows that the HWT admits a quantum realization using structured Hadamard gates, enabling its decomposition into unitary operations suitable for quantum circuits.

Abstract

Convolutional neural networks rely on linear filtering operations that can be reformulated efficiently in suitable transform domains. At the same time, advances in quantum computing have shown that certain structured linear transforms can be implemented with shallow quantum circuits, opening the door to hybrid quantum-classical approaches for enhancing deep learning models. In this work, we introduce WTHaar-Net, a convolutional neural network that replaces the Hadamard Transform used in prior hybrid architectures with the Haar Wavelet Transform (HWT). Unlike the Hadamard Transform, the Haar transform provides spatially localized, multi-resolution representations that align more closely with the inductive biases of vision tasks. We show that the HWT admits a quantum realization using structured Hadamard gates, enabling its decomposition into unitary operations suitable for quantum circuits. Experiments on CIFAR-10 and Tiny-ImageNet demonstrate that WTHaar-Net achieves substantial parameter reduction while maintaining competitive accuracy. On Tiny-ImageNet, our approach outperforms both ResNet and Hadamard-based baselines. We validate the quantum implementation on IBM Quantum cloud hardware, demonstrating compatibility with near-term quantum devices.
Paper Structure (32 sections, 33 equations, 5 figures, 3 tables)

This paper contains 32 sections, 33 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Haar transform.
  3. Hadamard transform.
  4. Hadamard gate.
  5. Related Work
  6. Wavelet-based CNNs.
  7. Hybrid quantum-classical neural networks.
  8. Quantum-friendly structured transforms.
  9. Preliminary
  10. 1D Haar Wavelet.
  11. Example.
  12. 2D Haar Wavelet.
  13. Example (2D Haar transform for $N=4$).
  14. Quantum circuit
  15. Methods
  16. ...and 17 more sections

Figures (5)

  • Figure 1: Illustration of the Haar wavelet filter bank. Approximation and detail coefficients are obtained by successive low-pass and high-pass filtering followed by downsampling.
  • Figure 2: A 4-dimensional input vector processed through the decomposed Haar circuit. Each coefficient $x_0, x_1$ corresponds to the amplitude of the quantum state in the computational basis, allowing the Hadamard gates to compute the wavelet transform directly on the quantum state.
  • Figure 3: Architecture of the Haar wavelet transform Conv Layer
  • Figure 4: Image corruptions used for robustness evaluation in the comparison between Haar (HWT) and Hadamard (WHT) architectures on Tiny-ImageNet.
  • Figure 5: Quantum circuit for the decomposition of the wavelet transform into Hadamard gates for a $4 \times 4$ patch.