"For an initial uniform superposition over all pos- sible computational basis states, we explore the perfor- mance of Grover’s search algorithm geometrically when imposing a perturbation on the Walsh-Hadamard trans- formation 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 per- turbation. Thereby we corroborate Grover’s conclusion that if the perturbation is small, then it will have little impact on the implementation of this algorithm. We also prove that Grover’s path cannot achieve a geodesic under a perturba- tion of the Fubini-Study metric.""For an initial uniform superposition over all pos- sible computational basis states, we explore the perfor- mance of Grover’s search algorithm geometrically when imposing a perturbation on the Walsh-Hadamard trans- formation 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 per- turbation. Thereby we corroborate Grover’s conclusion that if the perturbation is small, then it will have little impact on the implementation of this algorithm. We also prove that Grover’s path cannot achieve a geodesic under a perturba- tion of the Fubini-Study metric."

Jul 1, 20130 citations

Analysis of Grover’S Quantum Search Algorithm on a Classical Computer

Necati Çeli̇k, Özkan Bingöl

Jan 1, 20220 citations

Family of Grover’s quantum-searching algorithms

Alberto Galindo, Miguel Angel Martin-Delgado

We introduce the concepts of Grover operators and Grover kernels to systematically analyze Grover's searching algorithms. Then we investigate a one-parameter family of quantum searching algorithms of Grover type and we show that the standard Grover algorithm is a distinguished member of this family. We show that all the algorithms of this class solve the searching problem with an efficiency of order $O(\sqrt{N}),$ with a coefficient which is class-dependent. The analysis of this dependence is a test of the stability and robustness of the algorithms. We show the stability of this constructions under perturbations of the initial conditions and extend them to a very general class of Grover operators.We introduce the concepts of Grover operators and Grover kernels to systematically analyze Grover's searching algorithms. Then we investigate a one-parameter family of quantum searching algorithms of Grover type and we show that the standard Grover algorithm is a distinguished member of this family. We show that all the algorithms of this class solve the searching problem with an efficiency of order $O(\sqrt{N}),$ with a coefficient which is class-dependent. The analysis of this dependence is a test of the stability and robustness of the algorithms. We show the stability of this constructions under perturbations of the initial conditions and extend them to a very general class of Grover operators.

arXiv:quant-ph/0009086Physical Review ANov 8, 200050 citations

An improved formalism for the Grover search algorithm

James M. Chappell, Max A. Lohe, Lorenz von Smekal +2 more

The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is of significant interest to describe it in the most efficient mathematical formalism. We show firstly, that Clifford's formalism of geometric algebra, provides a significantly more efficient representation than the conventional Bra-ket notation, and secondly, that the basis defined by the states of maximum and minimum weight in the Grover search space, allows a simple visualization of the Grover search as the precession of a spin-1/2 particle. Using this formalism we efficiently solve the exact search problem, as well as easily representing more general search situations.The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is of significant interest to describe it in the most efficient mathematical formalism. We show firstly, that Clifford's formalism of geometric algebra, provides a significantly more efficient representation than the conventional Bra-ket notation, and secondly, that the basis defined by the states of maximum and minimum weight in the Grover search space, allows a simple visualization of the Grover search as the precession of a spin-1/2 particle. Using this formalism we efficiently solve the exact search problem, as well as easily representing more general search situations.

arXiv:1201.1707Jan 9, 20122 citations

O(\sqrt{N}),

with a coefficient which is class-dependent. The analysis of this dependence is a test of the stability and robustness of the algorithms. We show the stability of this constructions under perturbations of the initial conditions and extend them to a very general class of Grover operators.