fermions, the system’s thermodynamics is not expected to change with the dimensionality.

In all the situations we have addressed, we find a robust evidence of the fact that an optimal working time for the quantum annealing protocol emerges as a result of the competition between the non-adiabatic effects and the dissipative processes. There is an optimal time for which the final state is closest to the true final ground state. The scaling of such optimal time (and that of the corresponding generated defects) can be accurately predicted by assuming that the number of defects produced during the time evolution is a sum of the two contributions due to non-adiabaticity and dissipation/decoherence.

For larger working times it may happen that, depending on the type of system-bath coupling, an overshooting point sets in, where the density of the generated defects is larger than that for an infinitely slow annealing, which would adiabatically drive the system through the instantaneous steady state. While this kind of behaviour cannot appear in the unitary scenario, where the defects production is monotonic non-increasing with the annealing speed, in the system-bath scenario this can emerge even for small systems, being eventually related to the spectral structure of the 