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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 algo rithm This allows us to determine the number of iterations necessary to achieve almost certainty of nding the answer Furthermore we an alyze 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 nd such an element even when the number of solutions is not known ahead of time Finally we provide a lower bound on the e ciency of any possible quantum database searching algorithm and we show that Grover s algorithm comes within / of being optimal CHAPTER TIGHT BOUNDS ON QUANTUM SEARCHING Introduction LetXN f N g for some integerN and consider an arbitrary function F XN f g The goal is to nd some i XN such that F i provided such an i exists If F is given as a black box0the only knowledge you can gain about F is in asking for its value on arbitrary points of its domain0and if there is a unique solution no classical algorithm deterministic or probabilistic can expect to achieve a probability of success better than !/ without asking for the value of F on roughly N points Throughout this paper we assume for simplicity that each evaluation of F takes unit time Grover has discovered an algorithm for the quantum computer that can solve this problem in expected time in O p N provided there is a unique solution He also remarked that a result in implies that his algorithm is optimal up to an unspeci ed multiplicative constant among all possible quantum algorithms In this paper we provide a tight analysis of Grover s algorithm In particular we give a simple closed form formula for the probability of success after any given number of iterations This allows us to determine the number of iterations necessary to achieve almost certainty of nding the answer as well as an upper bound on the probability of failure More signi cantly we analyze the behaviour of the algorithm when there is an arbitrary number of solutions An algorithm follows immediately to solve the problem in a time in O p N t when it is known that there are exactly t solutions Moreover we provide an algorithm capable of solving the problem in a time in O p N t even if the number t of solutions is not known in advance We also generalize Grover s algorithm to the case N is not a power of Finally we re ne the argument of to show that Grover s algorithm is within / of being optimal To motivate this work here are three simple applications for Grover s algorithm Assume you have a large table T N in which you would like to nd some element y More precisely you wish to nd an integer i such that i N and T i y provided such an i exists This database searching problem can CHAPTER TIGHT BOUNDS ON QUANTUM SEARCHING obviously be solved in a time in O logN if the table is sorted but no classical algorithm can succeed in the general case with probability better than !/ say without probing more than half the entries of T Grover s algorithm solves this problem in a time in O p N on the quantum computer by using F i if and only if T i y An exciting cryptographic application is that Grover s algorithm can be used to crack the widely used Data Encryption Standard des under a known plaintext attack Given a matching pair m c of plaintext and cipher text consider function F f g f g de ned by F k if and only if desk m c Provided there is a unique solution the required key k can be found after roughly million expected calls to a quantum des device Thus quantum computing makes single key des totally insecure For yet another application consider a Boolean formula on n variables You would like to deter mine if the formula is satis able There may exist an e cient classical algorithm for this problem but none are known This is equivalent to the famous P  NP open question in theoretical computer science In this case Grover s algo rithm solves the problem in a time in O n which is better than the time in O n required by the obvious classical algorithm but not good enough to imply that NP BQP Overview of Grover s algorithm Grover s algorithm consists of an initialization followed by a number of identical iterations a nal measurement and a classical test For every F XN f g let SF be the conditional phase shift transform de ned by SF jii jii if F i jii otherwise Let S denote SF where F i if and only if i CHAPTER TIGHT BOUNDS ON QUANTUM SEARCHING Assume for the moment that N n is a power of and consider any integer j XN as a bit string of length n De ne i j as the number of in the bitwise and of i and j Let W be the Walsh-Hadamard transform de ned by