Geometric Analysis of Grover’s Search Algorithmin the Presence of Perturbation

Abstract

For an initial uniform superposition over all possible computational basis states, we explore the performance of Grover’s search algorithm geometrically when imposing a perturbation on the Walsh-Hadamard transformation contained in the Grover iteration. We give the geometric picture to visualize the quantum search process in the three-dimensional space and show that Grover’s search algorithm can work well with an appropriately chosen perturbation. Thereby we corroborate Grover’s conclusion that if the perturbation is small, then it will have little impact of an impact on the implementation of this algorithm. We also prove that Grover’s path cannot achieve a geodesic under a perturbation of the Fubini-Study metric.