Geometric analysis of Grover's search algorithm in 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 such perturbation is small, then this will not create much of an impact on the implementation of this algorithm. We also prove that Grover's path cannot achieve a geodesic in the presence of a perturbation of the Fubini-Study metric. PACS numbers: 02.70.-c, 03.67.-a, 03.65.-w, 03.67.Lx The Grover’s search algorithm [1] offers a quadratic speed-up over classical counterparts for searching an unsorted database, i.e., the problem of finding a desired state in an unsorted database of size N takes Θ