  We introduce an inhomogeneous protocol to drive a weakly disordered quantum
spin chain quasi-adiabatically across a quantum phase transition and minimize
the residual energy of the final state. The number of spins that simultaneously
reach the critical point is controlled by the length scale in which the
magnetic field is modulated, introducing an effective size that favors
adiabatic dynamics. The dependence of the residual energy on this length scale
and the velocity at which the magnetic field sweeps out the chain is shown to
be nonmonotonic. We determine the conditions for an optimal suppression of the
residual energy of the final state and show that inhomogeneous driving can
outperform conventional adiabatic schemes based on homogeneous control fields
by several orders of magnitude.

Quantum Physics;Statistical Mechanics