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Expressivity of quantum neural networks

Wu, Yadong and Yao, Juan and Zhang, Pengfei and Zhai, Hui (2021) Expressivity of quantum neural networks. Physical Review Research, 3 (3). Art. No. L032049. ISSN 2643-1564. doi:10.1103/PhysRevResearch.3.L032049.

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In this work, we address the question whether a sufficiently deep quantum neural network can approximate a target function as accurate as possible. We start with typical physical situations that the target functions are physical observables, and then we extend our discussion to situations that the learning targets are not directly physical observables, but can be expressed as physical observables in an enlarged Hilbert space with multiple replicas, such as the Loschmidt echo and the Rényi entropy. The main finding is that an accurate approximation is possible only when all the input wave functions in the dataset do not span the entire Hilbert space that the quantum circuit acts on, and more precisely, the Hilbert space dimension of the former has to be less than half of the Hilbert space dimension of the latter. In some cases, this requirement can be satisfied automatically because of the intrinsic properties of the dataset, for instance, when the input wave function has to be symmetric between different replicas. And if this requirement cannot be satisfied by the dataset, we show that the expressivity capabilities can be restored by adding one ancillary qubit at which the wave function is always fixed at input. Our studies point toward establishing a quantum neural network analogy of the universal approximation theorem that lays the foundation for expressivity of classical neural networks.

Item Type:Article
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URLURL TypeDescription Paper
Zhang, Pengfei0000-0002-7408-0918
Zhai, Hui0000-0001-8118-6027
Additional Information:© 2021 The Author(s). Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. (Received 24 April 2021; accepted 9 August 2021; published 23 August 2021) This work is supported by Beijing Outstanding Young Scientist Program (HZ), NSFC Grant No. 11734010 (HZ) and No. 11904190 (JY), MOST under Grant No. 2016YFA0301600 (HZ), and the Walter Burke Institute for Theoretical Physics at Caltech (PZ).
Group:Institute for Quantum Information and Matter, Walter Burke Institute for Theoretical Physics
Funding AgencyGrant Number
Beijing Outstanding Young Scientist ProgramUNSPECIFIED
National Natural Science Foundation of China11734010
National Natural Science Foundation of China11904190
Ministry of Science and Technology (China)2016YFA0301600
Walter Burke Institute for Theoretical Physics, CaltechUNSPECIFIED
Issue or Number:3
Record Number:CaltechAUTHORS:20210830-203806681
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110613
Deposited By: George Porter
Deposited On:31 Aug 2021 14:06
Last Modified:31 Aug 2021 14:06

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