Geometric pictures for quantum search algorithms

Abstract

In this paper, we give the geometric pictures for quantum search algorithms in decomposed form, namely in terms of a phase rotation of the marked state and a phase rotation about the average. We apply this formalism to various quantum search algorithms, and give explicit interpretations of the standard Grover algorithm, arbitrary phase rotations, phase matching and fixed-point search algorithm. The pictures straightforwardly show how state vectors evolve during the search process. These results are helpful in understanding how the quantum search algorithms work.