Quantum Alphatron: quantum advantage for learning with kernels and noise

Siyi Yang1, Naixu Guo1, Miklos Santha1,2, and Patrick Rebentrost1

1Centre for Quantum Technologies, National University of Singapore, Singapore 117543
2Université de Paris, IRIF, CNRS, F-75013 Paris, France; MajuLab UMI 3654

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Abstract

At the interface of machine learning and quantum computing, an important question is what distributions can be learned provably with optimal sample complexities and with quantum-accelerated time complexities. In the classical case, Klivans and Goel discussed the $Alphatron$, an algorithm to learn distributions related to kernelized regression, which they also applied to the learning of two-layer neural networks. In this work, we provide quantum versions of the Alphatron in the fault-tolerant setting. In a well-defined learning model, this quantum algorithm is able to provide a polynomial speedup for a large range of parameters of the underlying concept class. We discuss two types of speedups, one for evaluating the kernel matrix and one for evaluating the gradient in the stochastic gradient descent procedure. We also discuss the quantum advantage in the context of learning of two-layer neural networks. Our work contributes to the study of quantum learning with kernels and from samples.

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[2] João F. Doriguello, Alessandro Luongo, Jinge Bao, Patrick Rebentrost, and Miklos Santha, "Quantum algorithm for stochastic optimal stopping problems with applications in finance", arXiv:2111.15332, (2021).

[3] Debbie Lim and Patrick Rebentrost, "A quantum online portfolio optimization algorithm", Quantum Information Processing 23 3, 63 (2024).

[4] Yusen Wu, Bujiao Wu, Jingbo Wang, and Xiao Yuan, "Quantum Phase Recognition via Quantum Kernel Methods", arXiv:2111.07553, (2021).

[5] Yusen Wu, Bujiao Wu, Jingbo Wang, and Xiao Yuan, "Quantum Phase Recognition via Quantum Kernel Methods", Quantum 7, 981 (2023).

[6] Jeong Yu Han and Patrick Rebentrost, "Quantum advantage for multi-option portfolio pricing and valuation adjustments", arXiv:2203.04924, (2022).

[7] Armando Bellante and Stefano Zanero, "Quantum matching pursuit: A quantum algorithm for sparse representations", Physical Review A 105 2, 022414 (2022).

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