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Enabling Non-linear Quantum Operations Through Variational Quantum Splines

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Computational Science – ICCS 2023 (ICCS 2023)

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One of the major issues for building a complete quantum neural network is the implementation of non-linear activation functions in a quantum computer. In fact, the postulates of quantum mechanics impose only unitary transformations on quantum states, which is a severe limitation for quantum machine learning algorithms. Recently, the idea of QSplines has been proposed to approximate non-linear quantum activation functions by means of the HHL. However, QSplines rely on a problem formulation to be represented as a block diagonal matrix and need a fault-tolerant quantum computer to be correctly implemented.

This work proposes two novel methods for approximating non-linear quantum activation functions using variational quantum algorithms. Firstly, we develop the variational QSplines (VQSplines) that allow overcoming the highly demanding requirements of the original QSplines and approximating non-linear functions using near-term quantum computers. Secondly, we propose a novel formulation for QSplines, the Generalized QSplines (GQSplines), which provide a more flexible representation of the problem and are suitable to be embedded in existing quantum neural network architectures. As a third meaningful contribution, we implement VQSplines and GQSplines using Pennylane to show the effectiveness of the proposed approaches in approximating typical non-linear activation functions in a quantum computer.

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    It is also possible to define the Hermitian matrix H from S as \(H=\begin{pmatrix} 0 &{} S \\ S^\dag &{} 0 \end{pmatrix}\).

  2. 2.

    Though the primary objective of QSplines is to embed non-linearity into quantum neural networks, they can serve to approximate other types of non-linearity.

  3. 3.


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This work has been partially funded by the German Ministry for Education and Research (BMB+F) in the project QAI2-QAICO under grant 13N15586.

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Correspondence to Filippo Orazi .

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Inajetovic, M.A., Orazi, F., Macaluso, A., Lodi, S., Sartori, C. (2023). Enabling Non-linear Quantum Operations Through Variational Quantum Splines. In: Mikyška, J., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M. (eds) Computational Science – ICCS 2023. ICCS 2023. Lecture Notes in Computer Science, vol 14077. Springer, Cham.

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