Physics – Quantum Physics
Scientific paper
2009-03-03
Phys. Rev. A 81, 032324 (2010)
Physics
Quantum Physics
7 pages, 1 figure, published version
Scientific paper
10.1103/PhysRevA.81.032324
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 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.
Bisio Alessandro
Chiribella Giulio
D'Ariano Giacomo M.
Facchini Stefano
Perinotti Paolo
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