Lower bounds on the Deterministic and Quantum Communication Complexity of Hamming Distance

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 lower-bounding the ranks of the appropriate matrices.