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Abstract: We address the problem of learning an unknown unitary transformation from afinite number of examples. The problem consists in finding the learning machinethat optimally emulates the examples, thus reproducing the unknown unitarymaximum fidelity. Learning a unitary is equivalent to storing it in the stateof a quantum memory the memory of the learning machine, and subsequentlyretrieving it. We prove that, whenever the unknown unitary is drawn from agroup, the optimal strategy consists in a parallel call of the available usesfollowed by a -measure-and-rotate- retrieving. Differing from the case ofquantum cloning, where the incoherent -measure-and-prepare- strategies aretypically suboptimal, in the case of learning the -measure-and-rotate- strategyis optimal even when the learning machine is asked to reproduce a single copyof the unknown unitary. We finally address the problem of the optimal inversionof an unknown unitary evolution, showing also in this case the optimality ofthe -measure-and-rotate- strategies and applying our result to the optimalapproximate realignment of reference frames for quantum communication.

Author: A. Bisio, G. Chiribella, G. M. D'Ariano, S. Facchini, P. Perinotti

Source: https://arxiv.org/

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