Optimal quantum learning of a unitary transformation
Abstract
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 “measureandrotate” retrieving. Differing from the case of quantum cloning, where the incoherent “measureandprepare” strategies are typically suboptimal, in the case of learning the “measureandrotate” 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 “measureandrotate” strategies and applying our result to the optimal approximate realignment of reference frames for quantum communication.
 Publication:

Physical Review A
 Pub Date:
 March 2010
 DOI:
 10.1103/PhysRevA.81.032324
 arXiv:
 arXiv:0903.0543
 Bibcode:
 2010PhRvA..81c2324B
 Keywords:

 03.67.Lx;
 03.65.w;
 03.67.Hk;
 Quantum computation;
 Quantum mechanics;
 Quantum communication;
 Quantum Physics
 EPrint:
 7 pages, 1 figure, published version