Quantum and Classical Communication Complexity of Permutation-Invariant Functions

Ziyi Guan; Yunqi Huang; Penghui Yao; Zekun Ye

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arXiv · 2401.00454

Quantum and Classical Communication Complexity of Permutation-Invariant Functions

Ziyi Guan·Yunqi Huang·Penghui Yao·Zekun Ye

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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 \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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BibTeX RIS

Ziyi Guan, Yunqi Huang, Penghui Yao, Zekun Ye. 2023-12-31. Quantum and Classical Communication Complexity of Permutation-Invariant Functions. https://arxiv.org/abs/2401.00454

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