  We prove an adiabatic theorem that applies at timescales short of the
adiabatic limit. Our proof analyzes the stability of solutions to Schrodinger's
equation under perturbation. We directly characterize cross-subspace effects of
perturbation, which are typically significantly less than suggested by the
perturbation's operator norm. This stability has numerous consequences: we can
(1) find timescales where the solution of Schrodinger's equation converges to
the ground state of a block, (2) lower bound the convergence to the global
ground state by demonstrating convergence to some other known quantum state,
(3) guarantee faster convergence than the standard adiabatic theorem when the
ground state of the perturbed Hamiltonian ($H$) is close to that of the
unperturbed $H$, and (4) bound tunneling effects in terms of the global
spectral gap when $H$ is ``stoquastic'' (a $Z$-matrix). Our results apply to
quantum annealing protocols with faster convergence than usually guaranteed by
a standard adiabatic theorem. Our upper and lower bounds demonstrate that at
timescales short of the adiabatic limit, subspace dynamics can dominate over
global dynamics. Thus, we see that convergence to particular target states can
be understood as the result of otherwise local dynamics.
