Quasi-adiabatic Grover search via the WKB approximation

Abstract

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.