Query complexity of unitary operation discrimination

Discrimination of unitary operations is fundamental in quantum computation and information. A lot of quantum algorithms including the well-known Deutsch–Jozsa algorithm, Simon’s algorithm, and Grover’s algorithm can essentially be regarded as discriminating among individual, or sets of unitary operations (oracle operators). The problem of discriminating between two unitary operations U and V can be described as: Given X∈{U,V}, determine which one X is. If X is given with multiple copies, then one can design an adaptive procedure that takes multiple queries to X to output the identification result of X. In this paper, we consider the problem: How many queries are required for achieving a desired failure probability ϵ of discrimination between U and V. We prove in a uniform framework: (i) if U and V are discriminated with bound error ϵ , then the number of queries T must satisfy T≥21−4ϵ(1−ϵ)Θ(U†V), and (ii) if they are discriminated with one-sided error ϵ, then there is T≥21−ϵ2Θ(U†V), where ⌈k⌉ denotes the minimum integer not less than k and Θ(W) denotes the length of the smallest arc containing all the eigenvalues of W on the unit circle.