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      Tight bounds on quantum searching

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          Abstract

          We provide a tight analysis of Grover's recent 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. Using techniques from Shor's quantum factoring algorithm in addition to Grover's approach, we introduce a new technique for approximate quantum counting, which allows to estimate the number of solutions. Finally we provide a lower bound on the efficiency of any possible quantum database searching algorithm and we show that Grover's algorithm nearly comes within a factor 2 of being optimal in terms of the number of probes required in the table.

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          Rapid Solution of Problems by Quantum Computation

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            Author and article information

            Journal
            23 May 1996
            Article
            10.1002/(SICI)1521-3978(199806)46:4/5<493::AID-PROP493>3.0.CO;2-P
            quant-ph/9605034
            3d8eddd2-70b2-41ca-b2d5-7f508701494b
            Custom metadata
            Fortsch.Phys.46:493-506,1998
            8 pages, LaTeX2e, PhysComp 96 submission
            quant-ph

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