The communication complexity of the Hamming distance problem

We investigate the randomized and quantum communication complexity of the Hamming Distance problem, which is to determine if the Hamming distance between two n-bit strings is no less than a threshold d. We prove a quantum lower bound of Ω(d) qubits in the general interactive model with shared prior entanglement. We also construct a classical protocol of O(dlogd) bits in the restricted Simultaneous Message Passing model with public random coins, improving previous protocols of O(d2) bits [A.C.-C. Yao, On the power of quantum fingerprinting, in: Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 2003, pp. 77–81], and O(dlogn) bits [D. Gavinsky, J. Kempe, R. de Wolf, Quantum communication cannot simulate a public coin, quant-ph/0411051, 2004].