# Quasi-adiabatic Grover search via the WKB approximation

## TL;DR

This paper introduces a quasi-adiabatic WKB approximation for quantum dynamics at intermediate times, effectively capturing subtle features missed by traditional adiabatic methods, especially in Grover search problems.

## Contribution

It develops a novel WKB-based quasi-adiabatic approximation applicable to intermediate evolution times in quantum systems, improving analysis of quantum search algorithms.

## Key findings

- The method captures subtle dynamical features missed by adiabatic approximation.
- It reproduces the quadratic Grover speedup with the optimal schedule.
- Sensitivity to the interpolation schedule can lead to nonsensical results.

## 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.