Search for optimal driving in finite quantum systems with precursors of criticality

Using the adiabatic perturbation theory of driven dynamics [Phys. Rev. A 78, 052508 (2008)] we design a hierarchy of quantum state preparation protocols that systematically increase the fidelity at very long driving times. We test these and other protocols, including those based on the geometric analysis of the parameter space, in a single-qubit system and in a fully connected multi-qubit system showing in its infinite-size limit several quantum phase transitions. The new protocols excel in the asymptotic driving regime, above a crossover time from the Landau-Zener regime which increases with a decreasing minimal energy gap along the driving path (with the size of the system). In the medium-time domain, the performance of all tested protocols is indecisive.