Near-Optimal Bounds on the Bounded-Round Quantum Communication Complexity of Disjointness

Abstract

We prove a near optimal round-communication tradeoff for the two-party quantum communication complexity of disjointness. For protocols with r rounds, we prove a lower bound of Ω( ˜ n/r + r) on the communication required for computing disjointness of input size n, which is optimal up to logarithmic factors. The previous best lower bound was Ω(n/r2 + r) due to Jain, Radhakrishnan and Sen [Proceedings of FOCS, 2003, pp. 220–229]. Along the way, we develop several tools for quantum information complexity, one of which is a lower bound for quantum information complexity in terms of the generalized discrepancy method. As a corollary, we get that the quantum communication complexity of any Boolean function f is at most 2O(QIC(f)), where QIC(f) is the prior-free quantum information complexity of f (with error 1/3).