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

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

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