Promised and Distributed Quantum Search

This paper gives a quantum algorithm to search in an set S for a k-tuple satisfying some predefined relation, with the promise that some components of a desired k-tuple are in some subsets of S. In particular when k=2, we show a tight bound of the quantum query complexity for the Claw Finding problem, improving previous upper and lower bounds by Buhrman, Durr, Heiligman, Hoyer, Magniez, Santha and de Wolf [7]. We also consider the distributed scenario, where two parties each holds an n-element set, and they want to decide whether the two sets share a common element. We show a family of protocols s.t.q(P)3/2 . c(P)= O(n2log n), where q(P) and c(P) are the number of quantum queries and the number of communication qubits that the protocol P makes, respectively. This implies that we can pay more for quantum queries to save on quantum communication, and vice versa. To our knowledge, it is the first result about the tradeoff between the two resources.