Lower Bounds on the Deterministic and Quantum Communication Complexity of Hamming-Distance Problems

Abstract

Alice and Bob want to know if two strings of length n are almost equal. That is, do the strings differ on at most a bits? Let 0 ⩽ a ⩽ n − 1. We show (1) any deterministic protocol—as well as any error-free quantum protocol ( C * version)—for this problem requires at least n − 2 bits of communication, and (2) a lower bound of n /2 − 1 for error-free Q * quantum protocols. We also show the same results for determining if two strings differ in exactly a bits. Our results are obtained by lower-bounding the ranks of the appropriate matrices.