Optimal quantum learning of a unitary transformation

TL;DR

This paper investigates the optimal methods for learning and reproducing unknown unitary transformations in quantum systems, demonstrating that measure-and-rotate strategies are optimal for various tasks including inversion and reference frame realignment.

Contribution

It proves the optimality of measure-and-rotate strategies for learning and inverting unknown unitaries, extending understanding beyond quantum cloning scenarios.

Findings

Optimal strategy involves parallel calls followed by measure-and-rotate.

Measure-and-rotate is optimal even for reproducing a single copy.

Application to optimal realignment of quantum reference frames.

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 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.