Optimal processing of reversible quantum channels

• We study the problem of transforming N unitary channels into another unitary channel.
• This class of problems includes the cloning of quantum gates as a special case.
• The task can be fully parallelized and its degrees of freedom exponentially reduced.
• Some special cases, like the cloning of qubit phase gates, are studied in detail.

We consider the general problem of the optimal transformation of N uses of (possibly different) unitary channels to a single use of another unitary channel in any finite dimension. We show how the optimal transformation can be fully parallelized, consisting in a preprocessing channel followed by a parallel action of all the N unitaries and a final postprocessing channel. Our techniques allow to achieve an exponential reduction in the number of the free parameters of the optimization problem making it amenable to an efficient numerical treatment. Finally, we apply our general results to find the analytical solution for special cases of interest like the cloning of qubit phase gates.