Quantum and Classical Communication Complexity of Permutation-Invariant Functions

TL;DR

This work analyzes quantum versus randomized two-party communication complexity for permutation-invariant Boolean functions, showing that symmetry prevents exponential quantum speedups and enabling a near-tight relationship between $R(f)$ and $Q(f)$ up to polylog factors. It introduces the measure $m(f)$ to capture PI complexity and proves $Q(f) = \Omega(m(f))$ and $R(f) = O(m(f)^2 \log^2 n \log\log n + \log n)$, with $Q(f) = O(m(f) \log^2 n \log\log n + \log n)$, yielding a quasi-quadratic gap $R(f) \le O(Q(f)^2 \log^2 n \log\log n)$. The authors also establish Log-rank and Log-approximate-rank conjectures for non-trivial permutation-invariant functions, showing that deterministic complexity and (approximate) rank-based bounds are polylogarithmic in the input size. By combining reductions to the Exact Set-Inclusion problem with pattern-matrix methods and amplitude amplification, the paper extends symmetry-based quantum limitations from query to communication complexity, highlighting the fundamental role of structure in limiting quantum speedups. These results provide a unified framework for understanding the limits of quantum advantages under permutation symmetry and open directions to higher alphabets and broader symmetry classes.

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).

Figures (1)

• Figure 1: Our reduction path for the lower bound of $\mathsf{ESetInc}$.

Theorems & Definitions (73)

• Definition 1.1: Permutation-invariant (PI) functions GKS16
• Remark 1.2
• Conjecture 1.3: Communication complexity version of the Watrous Conjecture
• Conjecture 1.4: Log-rank Conjecture for permutation-invariant functions
• Conjecture 1.5: Log-Approximate-Rank Conjecture for permutation-invariant functions
• Conjecture 1.6: Quantum Log-Approximate-Rank Conjecture for permutation-invariant functions
• Definition 1.7: Approximate rank
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10