Quantum query complexity of unitary operator discrimination

Abstract

Unitary operator discrimination is a fundamental problem in quantum information theory. The basic version of this problem can be described as follows: Given a black box implementing a unitary operator U ∈ S := {U 1 , U 2 } under some probability distribution over S, the goal is to decide whether U = U 1 or U = U 2 . In this paper, we consider the query complexity of this problem. We show that there exists a quantum algorithm that solves this problem with bounded error probability using 6θ cover−1 queries to the black box in the worst case, i.e., under any probability distribution over S, where the parameter θ cover , which is determined by the eigenvalues of U 1† U 2 , represents the “closeness” between U 1 and U 2 . We also show that this upper bound is essentially tight: we prove that for every θ cover > 0 there exist operators U 1 and U 2 such that any quantum algorithm solving this problem with bounded error probability requires at least 3θcover2 queries under uniform distribution over S.