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 quantum operator U, and the promise that the black box implements either the unitary operator $U_1$ or the unitary operator $U_2$, 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 $\left\lceil \frac{\pi }{3\theta _\mathrm{cover}} \right\rceil$ queries to the black-box, where $\theta _\mathrm{cover}$ represents the “closeness” between $U_1$ and $U_2$ (this parameter is determined by the eigenvalues of the matrix $U_1^\dag U_2$). We also show that this upper bound is essentially tight: we prove that there exist operators $U_1$ and $U_2$ such that any quantum algorithm solving this problem with bounded-error probability requires at least $\left\lceil \frac{2}{3\theta _\mathrm{cover}} \right\rceil$ queries.