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Quantum and Classical Communication Complexity of Permutation-Invariant Functions (2401.00454v1)

Published 31 Dec 2023 in cs.CC and quant-ph

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 \cite{AA14, Cha19, BCG+20} 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).

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Authors (4)
  1. Ziyi Guan (4 papers)
  2. Yunqi Huang (6 papers)
  3. Penghui Yao (42 papers)
  4. Zekun Ye (14 papers)

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