  We consider the optimal driving of the ground state of a many-body quantum
system across a quantum phase transition in finite time. In this context,
excitations caused by the breakdown of adiabaticity can be minimized by
adjusting the schedule of the control parameter that drives the transition.
Drawing inspiration from the Kibble-Zurek mechanism, we characterize the
timescale of onset of adiabaticity for several optimal control procedures. Our
analysis reveals that schedules relying on local adiabaticity, such as
Roland-Cerf's local adiabatic driving and the quantum adiabatic
brachistochrone, fail to provide a significant speedup over the adiabatic
evolution in the transverse-field Ising and long-range Kitaev models. As an
alternative, we introduce a novel framework, Non-Adiabatic Quantum Optimization
(NAQO), that, by exploiting the Landau-Zener formula and taking into account
the role of higher-excited states, outperforms schedules obtained via both
local adiabaticity and state-of-the-art numerical optimization. NAQO is not
restricted to exactly solvable models, and we further confirm its superior
performance in a disordered non-integrable model.
