Quantum Query Complexity Of Unitary Operator Discrimination

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 is an element of 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 inverted right perpendicular root 6 theta(-1)(cover) inverted left perpendicular queries to the black box in the worst case, i.e., under any probability distribution over S, where the parameter theta(cover), which is determined by the eigenvalues of U dagger U-1(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 theta(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 inverted right perpendicular 2/3 theta cover inverted left perpendicular queries under uniform distribution over S.