## Tight bounds on quantum searching (1996)

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Citations: | 114 - 10 self |

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@INPROCEEDINGS{Boyer96tightbounds,

author = {Michel Boyer and Université De Montréal and Gilles Brassard and Université De Montréal and Université De Montréal and Peter Høyer},

title = {Tight bounds on quantum searching},

booktitle = {},

year = {1996},

pages = {36--43}

}

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### Abstract

We provide a tight analysis of Grover’s algorithm for quantum database searching. We give a simple closed-form formula for the probability of success after any given number of iterations of the algorithm. This allows us to determine the number of iterations necessary to achieve almost certainty of finding the answer. Furthermore, we analyse the behaviour of the algorithm when the element to be found appears more than once in the table and we provide a new algorithm to find such an element even when the number of solutions is not known ahead of time. Finally, we provide a lower bound on the efficiency of any possible quantum database searching algorithm and we show that Grover’s algorithm comes within 2.62 % of being optimal.

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Citation Context ...usvej 55, DK-5230 Odense M, Denmark; u2pi@imada.ou.dk. k ) Supported in part by a postgraduate fellowship from Canada's nserc.494 M. Boyer, et al., Tight Bounds on Quantum Searching that a result in =-=[2]-=- implies that his algorithm is optimal, up to an unspecified multiplicative constant, among all possible quantum algorithms. In this paper we provide a tight analysis of Grover's algorithm. In particu... |

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Citation Context ...: f0; 1g 56 !f0; 1g defined by F…k† ˆ1 if and only if desk…m† ˆc. Provided there is a unique solution, the required key k can be found after roughly 185 million expected calls to a quantum des device =-=[3]-=-. Thus quantum computing makes single-key des totally insecure. For yet another application, consider a Boolean formula on n variables. You would like to determine if the formula is satisfiable. There... |

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Citation Context ...number of 1 in the bitwise and of i and j. Let W be the Walsh-Hadamard transform defined by W jji ˆ 1 p N P N 1 i ˆ 0 … 1† i j jii :Fortschr. Phys. 46 (1998) 4 ±5 495 This is efficiently implemented =-=[5]-=- by applying the simple unitary transformation 1 1 1 p independently to each qubit of jji. Now we can define Grover's iteration as 2 1 1 the unitary transformation GF ˆ WS0WSF : …1† Grover's algorithm... |

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