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  • Letter
  • Open Access

Expressivity of quantum neural networks

Yadong Wu1, Juan Yao2, Pengfei Zhang3,4,*, and Hui Zhai1,†

  • 1Institute for Advanced Study, Tsinghua University, Beijing, 100084, China
  • 2Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, Guangdong, China
  • 3Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
  • 4Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA

  • *pengfeizhang.physics@gmail.com
  • †hzhai@tsinghua.edu.cn
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Phys. Rev. Research 3, L032049 – Published 23 August, 2021

DOI: https://doi.org/10.1103/PhysRevResearch.3.L032049

Abstract

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.

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Phys. Rev. Research 3, L032049– Published 23 August, 2021
  • Received 24 April 2021
  • Accepted 9 August 2021
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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.

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