  In various applications one is interested in quantum dynamics at intermediate
evolution times, for which the adiabatic approximation is inadequate. Here we
develop a quasi-adiabatic approximation based on the WKB method, designed to
work for such intermediate evolution times. We apply it to the problem of a
single qubit in a time-varying magnetic field, and to the Hamiltonian Grover
search problem, and show that already at first order, the quasi-adiabatic WKB
captures subtle features of the dynamics that are missed by the adiabatic
approximation. However, we also find that the method is sensitive to the type
of interpolation schedule used in the Grover problem, and can give rise to
nonsensical results for the wrong schedule, due to loss of normalization.
Conversely, it reproduces the quadratic Grover speedup when the well-known
optimal schedule is used.
