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Manipulating quantum systems undergoing non-Gaussian dynamics in a fast andaccurate manner is becoming fundamental to many quantum applications. Here, wefocus on classical and quantum protocols transferring a state across adouble-well potential. The classical protocols are achieved by deforming thepotential, while the quantum ones are assisted by a counter-diabatic driving.We show that quantum protocols perform more quickly and accurately. Finally, wedesign a figure of merit for the performance of the transfer protocols --namely, the \textit{protocol grading} -- that depends only on fundamentalphysical quantities, and which accounts for the quantum speed limit, thefidelity and the thermodynamic of the process. We test the protocol gradingwith classical and quantum protocols, and show that quantum protocols havehigher protocol grading than the classical ones.

Quantifying protocol efficiency: a thermodynamic figure of merit for classical and quantum state-transfer protocols