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Quantum and Classical Communication Complexity of Permutation-Invariant Functions

Authors Ziyi Guan, Yunqi Huang, Penghui Yao, Zekun Ye

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  • Theory of computation → Models of computation
Keywords
  • Communication complexity
  • Permutation-invariant functions
  • Log-rank Conjecture
  • Quantum advantages

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Abstract

This paper gives a nearly tight characterization of the quantum communication complexity of the permutation-invariant Boolean functions. With such a characterization, we show that the quantum and randomized communication complexity of the permutation-invariant Boolean functions are quadratically equivalent (up to a logarithmic factor). Our results extend a recent line of research regarding query complexity [Scott Aaronson and Andris Ambainis, 2014; André Chailloux, 2019; Shalev Ben-David et al., 2020] to communication complexity, showing symmetry prevents exponential quantum speedups.
Furthermore, we show the Log-rank Conjecture holds for any non-trivial total permutation-invariant Boolean function. Moreover, we establish a relationship between the quantum/classical communication complexity and the approximate rank of permutation-invariant Boolean functions. This implies the correctness of the Log-approximate-rank Conjecture for permutation-invariant Boolean functions in both randomized and quantum settings (up to a logarithmic factor).

Cite As Get BibTex

Ziyi Guan, Yunqi Huang, Penghui Yao, and Zekun Ye. Quantum and Classical Communication Complexity of Permutation-Invariant Functions. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 39:1-39:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.STACS.2024.39

Author Details

Ziyi Guan
  • EPFL, Lausanne, Switzerland
Yunqi Huang
  • University of Technology Sydney, Australia
Penghui Yao
  • State Key Laboratory for Novel Software Technology, Nanjing University, China
  • Hefei National Laboratory, China
Zekun Ye
  • State Key Laboratory for Novel Software Technology, Nanjing University, China

Funding

  • Guan, Ziyi: partially supported by the Ethereum Foundation.
  • Yao, Penghui: supported by National Natural Science Foundation of China (Grant No. 62332009, 12347104, 61972191) and Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302901).
  • Ye, Zekun: supported by National Natural Science Foundation of China (Grant No. 62332009, 12347104, 61972191) and Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302901).

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