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:={U₁, U₂} under some probability distribution over S, the goal is to decide whether U=U₁ or U=U₂. 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 $\lceil{\sqrt{6}\theta_{\rm cover}^{-1}}\rceil$ 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 $U_1^\dagger {U_2}$, represents the “closeness” between U₁ and U₂. We also show that this upper bound is essentially tight: we prove that for every θ_cover > 0 there exist operators U₁ and U₂ such that any quantum algorithm solving this problem with bounded error probability requires at least $\lceil{\frac{2}{3 \theta_{\rm cover}}}\rceil$ queries under uniform distribution over S.