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).

翻译：本文给出了置换不变布尔函数量子通信复杂度的近乎紧致刻画。基于该刻画，我们证明了置换不变布尔函数的量子与随机化通信复杂度呈二次等价关系（仅相差对数因子）。我们的结果将近期关于查询复杂度的一系列研究（\cite{AA14, Cha19, BCG+20}）拓展至通信复杂度领域，表明对称性阻止了指数量子加速。此外，我们证明了对任意非平凡全置换不变布尔函数，对数秩猜想均成立。进一步地，我们建立了置换不变布尔函数的量子/经典通信复杂度与其近似秩之间的关联。这暗示了在随机化和量子两种设定下（均仅相差对数因子），置换不变布尔函数的对数近似秩猜想成立。