  Manipulating quantum systems undergoing non-Gaussian dynamics in a fast and
accurate manner is becoming fundamental to many quantum applications. Here, we
focus on classical and quantum protocols transferring a state across a
double-well potential. The classical protocols are achieved by deforming the
potential, while the quantum ones are assisted by a counter-diabatic driving.
We show that quantum protocols perform more quickly and accurately. Finally, we
design a figure of merit for the performance of the transfer protocols --
namely, the \textit{protocol grading} -- that depends only on fundamental
physical quantities, and which accounts for the quantum speed limit, the
fidelity and the thermodynamic of the process. We test the protocol grading
with classical and quantum protocols, and show that quantum protocols have
higher protocol grading than the classical ones.
