Lower bounds on the deterministic and quantum communication complexities of hamming-distance problems

Abstract

Alice and Bob want to know if two strings of length n are almost equal. That is, do they differ on at most a bits? Let 0≤ a≤n-1. We show that any deterministic protocol, as well as any error-free quantum protocol (C * version), for this problem requires at least n-2 bits of communication. We show the same bounds for the problem of determining if two strings differ in exactly a bits. We also prove a lower bound of n/2-1 for error-free Q* quantum protocols. Our results are obtained by employing basic tools from combinatorics and calculus to lower-bound the ranks of the appropriate matrices. © 2006 Springer-Verlag Berlin/Heidelberg.