Quantum Lazy Training

Erfan Abedi, Salman Beigi, and Leila Taghavi

QuOne Lab, Phanous Research & Innovation Centre, Tehran, Iran

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In the training of over-parameterized model functions via gradient descent, sometimes the parameters do not change significantly and remain close to their initial values. This phenomenon is called $\textit{lazy training}$ and motivates consideration of the linear approximation of the model function around the initial parameters. In the lazy regime, this linear approximation imitates the behavior of the parameterized function whose associated kernel, called the $\textit{tangent kernel}$, specifies the training performance of the model. Lazy training is known to occur in the case of (classical) neural networks with large widths. In this paper, we show that the training of $\textit{geometrically local}$ parameterized quantum circuits enters the lazy regime for large numbers of qubits. More precisely, we prove bounds on the rate of changes of the parameters of such a geometrically local parameterized quantum circuit in the training process, and on the precision of the linear approximation of the associated quantum model function; both of these bounds tend to zero as the number of qubits grows. We support our analytic results with numerical simulations.

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Cited by

[1] Massimiliano Incudini, Michele Grossi, Antonio Mandarino, Sofia Vallecorsa, Alessandra Di Pierro, and David Windridge, "The Quantum Path Kernel: A Generalized Neural Tangent Kernel for Deep Quantum Machine Learning", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[2] Junyu Liu, Zexi Lin, and Liang Jiang, "Laziness, barren plateau, and noises in machine learning", Machine Learning: Science and Technology 5 1, 015058 (2024).

[3] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, and Dacheng Tao, "Erratum: Learnability of Quantum Neural Networks [PRX QUANTUM 2, 040337 (2021)]", PRX Quantum 3 3, 030901 (2022).

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