Optimal quantum learning of a unitary transformation

We address the problem of learning an unknown unitary transformation from a finite number of examples. The problem consists in finding the learning machine that optimally emulates the examples, thus reproducing the unknown unitary with maximum fidelity. Learning a unitary is equivalent to storing it in the state of a quantum memory (the memory of the learning machine) and subsequently retrieving it. We prove that, whenever the unknown unitary is drawn from a group, the optimal strategy consists in a parallel call of the available uses followed by a ``measure-and-rotate'' retrieving. Differing from the case of quantum cloning, where the incoherent ``measure-and-prepare'' strategies are typically suboptimal, in the case of learning the ``measure-and-rotate'' strategy is optimal even when the learning machine is asked to reproduce a single copy of the unknown unitary. We finally address the problem of the optimal inversion of an unknown unitary evolution, showing also in this case the optimality of the ``measure-and-rotate'' strategies and applying our result to the optimal approximate realignment of reference frames for quantum communication.