arXiv:quant-ph/0009086 · Physical Review A

Family of Grover’s quantum-searching algorithms

Alberto Galindo, Miguel Angel Martin-Delgado

Published Nov 8, 2000• 50 citations

Abstract

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.

DOI: 10.1103/physreva.62.062303

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arXiv:quant-ph/0009086 · Physical Review A

Family of Grover’s quantum-searching algorithms

Alberto Galindo, Miguel Angel Martin-Delgado

Published Nov 8, 2000• 50 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physreva.62.062303

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arXiv:quant-ph/0210201International Journal of Quantum InformationAug 30, 20042 citations

Quantum Search Algorithm

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This work considers a generalization of Grover's search problem, viz., to find any one element in a set of acceptable choices which constitute a fraction f of the total number of choices in an unsorted database. An infinite family of sure-success quantum algorithms are introduced here to solve this problem, each member for a different range of f. The nth member of this family involves n queries of the database, and so the lowest few members of this family should be very convenient algorithms within their ranges of validity. The even member ${\mathcal{A}}_{2n}$ of the family covers an ever larger range of f for larger n, which is expected to become the full range $0<~f<~1$ in the limit $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}.$ These algorithms are particularly useful when the cost of failure of a search is very high, and for multistage searches with a different search criterion for each stage.This work considers a generalization of Grover's search problem, viz., to find any one element in a set of acceptable choices which constitute a fraction f of the total number of choices in an unsorted database. An infinite family of sure-success quantum algorithms are introduced here to solve this problem, each member for a different range of f. The nth member of this family involves n queries of the database, and so the lowest few members of this family should be very convenient algorithms within their ranges of validity. The even member ${\mathcal{A}}_{2n}$ of the family covers an ever larger range of f for larger n, which is expected to become the full range $0<~f<~1$ in the limit $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}.$ These algorithms are particularly useful when the cost of failure of a search is very high, and for multistage searches with a different search criterion for each stage.

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Apr 1, 20201 citation

O(\sqrt{N}),

{\mathcal{A}}_{2n}

of the family covers an ever larger range of f for larger n, which is expected to become the full range

0<~f<~1

in the limit $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}.$ These algorithms are particularly useful when the cost of failure of a search is very high, and for multistage searches with a different search criterion for each stage.

{I}_{s}

, the selective phase inversion of the initial state, also known as a diffusion operator. This assumption is known to become a serious handicap in cases of physical interest. Our general search algorithm works with almost any diffusion operator

{D}_{s}

with the only restriction of having $|s\ensuremath{\rangle}$ as one of its eigenstates. The price that we pay for using any operator is an increase in the number of oracle queries by a factor of

O(B)

, where

B

is a characteristic of the eigenspectrum of

{D}_{s}

and can be large in some situations. Here we show that by using a quantum Fourier transform, we can regain the optimal query complexity of Grover's algorithm without losing the freedom of using any diffusion operator for quantum searching. However, the total number of operators required by the algorithm is still

O(B)

times more than that of Grover's algorithm. So our algorithm offers an advantage only if the oracle operator is computationally more expensive than the diffusion operator, which is true in most search problems.