Comparison of unknown unitary channels with multiple uses

Comparison of quantum objects is a task to determine whether two unknown quantum objects are the same or different. It is one of the most basic information processing tasks for learning property of quantum objects, and comparison of quantum states, quantum channels, and quantum measurements have been investigated. In general, repeated uses of quantum objects improve the success probability of comparison. The optimal strategy of pure-state comparison, the comparison of quantum states for the case of multiple copies of each unknown pure state, is known, but the optimal strategy of unitary comparison, the comparison of quantum channels for the case of multiple uses of each unknown unitary channel, was not known due to the complication of the varieties of causal order structures among the uses of each unitary channel. In this paper, we investigate unitary comparison with multiple uses of unitary channels based on the quantum tester formalism. We obtain the optimal minimum-error and the optimal unambiguous strategies of unitary comparison of two unknown $d$-dimensional unitary channels $U_1$ and $U_2$ when $U_1$ can be used $N_1$ times and $U_2$ can be used $N_2$ times for $N_2 \ge (d-1)N_1$. These optimal strategies are implemented by parallel uses of the unitary channels, even though all sequential and adaptive strategies implementable by the quantum circuit model are considered. When the number of the smaller uses of the unitary channels $N_1$ is fixed, the optimal averaged success probability cannot be improved by adding more uses of $U_2$ than $N_2 = (d-1) N_1$. This feature is in contrast to the case of pure-state comparison, where adding more copies of the unknown pure states always improves the optimal averaged success probability. It highlights the difference between corresponding tasks for states and channels, which has been previously shown for quantum discrimination tasks.