  The quantum Ising chain has shortcuts to adiabaticity when operated with weak
processes. However, when exactly do the non-equilibrium effects of the
Kibble-Zurek mechanism, inherent to the system, appear in the optimal protocols
in such a context? I propose here that such contrasting difference occurs due
to the manner by which one measures the excitation spent energy of the system.
Therefore, in this work, I made a qualitative analysis of a quantum annealing
procedure of the time-averaged excess work, where the system acquires as a
diverging decorrelation time the heuristic Kibble-Zurek mechanism relaxation
time. Four important effects are then observed: the absence of shortcuts to
adiabaticity, the pausing effect around the critical point in the optimal
protocol when the Kibble-Zurek mechanism holds, the persistence of the
time-averaged work to avoid slowly-varying regime even for large switching
times, and diverging fluctuations of the time-averaged work. In the end, by
comparing the excess and the time-averaged excess works, I conclude that this
last one is not useful to measure the excitation spent energy in weak
processes, although brings an intuition to what happens in the strong driving
case.
