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arXiv:2206.01808 · Advances in Applied Clifford Algebras

Clifford Algebras, Quantum Neural Networks and Generalized Quantum Fourier Transform

Marco A. S. Trindade, Vinícius N. A. Lula-Rocha, Sérgio Floquet

Published Jun 13, 2023• 2 citations
arXivHTMLPDF
DOI
10.1007/s00006-023-01279-7

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A hybrid model called thegeometric (Clifford) quanvolutional neural network(GQNN) that merges elements of geometric (Clifford) convolutional neural networks (GCNNs) and variational quantum circuits (VQCs) is presented. In this model, a randomized quantum convolution operation is applied to the input image, giving as a result four output channels, which are treated as a single entity (quaternion image) by the subsequent quaternion layers. This approach is extended to Clifford algebras by choosing the number of qubits of the quantum circuit according to the dimension of the Clifford algebra so that the resulting output channels are regarded as the components of a multivector image to be further processed by Clifford layers.
Quantum neural networks (QNNs), typically implemented via parameterized quantum circuits (PQCs), offer a potential route to quantum advantage in learning. We review recent progress through the Fourier structure of PQCs: the encoding Hamiltonian fixes the accessible frequency set, while training estimates the corresponding Fourier coefficients. This perspective explains how QNNs can mitigate the spectral bias that limits classical deep neural networks and motivates task-specialized designs. In particular, quantum ordinary differential equation (QODE) solver, quantum convolutional neural networks (QCNNs) and quantum reinforcement learning (QRL) exploit frequency pinching to better capture high-frequency image features and structured value landscapes. We further discuss trainability on noisy intermediate-scale quantum (NISQ) hardware, emphasizing Lie-algebraic and Fourier views of barren plateaus and practical mitigation strategies. Together, the Fourier and Lie perspectives yield concrete design principles and motivate quantum-native architectures built around spectral bias avoidance rather than classical imitation. Quantum neural networks (QNNs), implemented via parameterized quantum circuits (PQCs), promise a path to quantum advantage in learning by addressing the limitations of classical deep neural networks. Here, the authors explore the Fourier structure of PQCs to mitigate spectral bias, enhancing applications like quantum ODE solvers and quantum reinforcement learning, with implications for designing quantum-native architectures.
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