Statistical complexity of quantum circuits

Kaifeng Bu, Dax Enshan Koh, Lu Li, Qingxian Luo, and Yaobo Zhang
Phys. Rev. A 105, 062431 – Published 17 June 2022

Abstract

In theoretical machine learning, the statistical complexity is a notion that measures the richness of a hypothesis space. In this work, we apply a particular measure of statistical complexity, namely, the Rademacher complexity, to the quantum circuit model in quantum computation and study how the statistical complexity depends on various quantum circuit parameters. In particular, we investigate the dependence of the statistical complexity on the resources, depth, width, and the number of input and output registers of a quantum circuit. To study how the statistical complexity scales with resources in the circuit, we introduce a magic resource measure based on the (p,q) group norm, which quantifies the amount of magic resource in the quantum channels associated with the circuit. These dependencies are investigated in the following two settings: (i) where the entire quantum circuit is treated as a single quantum channel, and (ii) where each layer of the quantum circuit is treated as a separate quantum channel. The bounds we obtain can be used to constrain the capacity of quantum neural networks in terms of their depths and widths as well as the resources in the network.

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  • Received 12 December 2021
  • Accepted 11 May 2022

DOI:https://doi.org/10.1103/PhysRevA.105.062431

©2022 American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & Technology

Authors & Affiliations

Kaifeng Bu1,*, Dax Enshan Koh2,†, Lu Li3,4,‡, Qingxian Luo4,5,§, and Yaobo Zhang6,7,∥

  • 1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
  • 2Institute of High Performance Computing, Agency for Science, Technology and Research (A*STAR), 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore
  • 3Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China
  • 4School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, China
  • 5Center for Data Science, Zhejiang University, Hangzhou, Zhejiang 310027, China
  • 6Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China
  • 7Department of Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China

  • *kfbu@fas.harvard.edu
  • dax_koh@ihpc.a-star.edu.sg
  • lilu93@zju.edu.cn
  • §luoqingxian@zju.edu.cn
  • yaobozhang@zju.edu.cn

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Vol. 105, Iss. 6 — June 2022

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