  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 $X\in\{U, V\}$,
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 $\epsilon$ of
discrimination between $U$ and $V$. We prove in a uniform framework: (i) if $U$
and $V$ are discriminated with bound error $\epsilon$ , then the number of
queries $T$ must satisfy $T\geq
\left\lceil\frac{2\sqrt{1-4\epsilon(1-\epsilon)}}{\Theta (U^\dagger
V)}\right\rceil$, and (ii) if they are discriminated with one-sided error
$\epsilon$, then there is $T\geq \left\lceil\frac{2\sqrt{1-\epsilon^2}}{\Theta
(U^\dagger V)}\right\rceil$, where $\lceil k\rceil$ denotes the minimum integer
not less than $k$ and $\Theta(W)$ denotes the length of the smallest arc
containing all the eigenvalues of $W$ on the unit circle.
