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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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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).

Cite AsGet 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

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).

Subject Classification

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

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