Journal
PHYSICAL REVIEW A
Volume 81, Issue 3, Pages -Publisher
AMER PHYSICAL SOC
DOI: 10.1103/PhysRevA.81.032324
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Funding
- Italian Ministry of Education [PRIN 2008]
- EC
- Risk and Security Study Center of IUSS Pavia
- Government of Canada through NSERC
- Ontario through MRI
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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.
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