On the stability of solutions to Schrödinger's equation short of the adiabatic limit

Abstract: We prove an adiabatic theorem that applies at timescales short of the typical 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 subspace, (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 is close to that of the unperturbed Hamiltonian, and (4) bound leakage effects in terms of the global spectral gap when the Hamiltonian is ``stoquastic'' (a $Z$-matrix). Our results can help explain quantum annealing protocols that exhibit faster convergence than is 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, when our results apply, convergence to particular global target states can be understood as the result of local dynamics.