An unsorted database contains N records, of which just one satisfies a particular property.The problem is to identify that one record.Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records.Quantum mechanical systems can do several operations simultaneously due to their wave like properties.This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record.It is within a constant factor of the fastest possible quantum mechanical algorithm.
An unsorted database contains N records, of which just one satisfies a particular property.The problem is to identify that one record.Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records.Quantum mechanical systems can do several operations simultaneously due to their wave like properties.This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record.It is within a constant factor of the fastest possible quantum mechanical algorithm.
Imagine a phone directory containing N names arranged in completely random order. In order to find someone's phone number with a 50% probability, any classical algorithm (whether deterministic or probabilistic) will need to look at a minimum of N/2 names. Quantum mechanical systems can be in a superposition of states and simultaneously examine multiple names. By properly adjusting the phases of various operations, successful computations reinforce each other while others interfere randomly. As a result, the desired phone number can be obtained in only O(sqrt(N)) steps. The algorithm is within a small constant factor of the fastest possible quantum mechanical algorithm.
We consider the problem of finding one or more desired items out of an unsorted database. Patel has shown that if the database permits quantum queries, then mere digitization is sufficient for efficient search for one desired item. The algorithm, called factorized quantum search algorithm, presented by him can locate the desired item in an unsorted database using O(log4N) queries to factorized oracles. But the algorithm requires that all the property values must be distinct from each other. In this paper, we discuss how to make a database satisfy the requirements, and present a quantum search engine based on the algorithm. Our goal is achieved by introducing auxiliary files for the property values that are not distinct, and converting every complex query request into a sequence of calls to factorized quantum search algorithm. The query complexity of our algorithm is O(P∗Q∗M∗log4N), where P is the number of the potential simple query requests in the complex query request, Q is the maximum number of calls to the factorized quantum search algorithm of the simple queries, M is the number of the auxiliary files for the property on which our algorithm are searching for desired items. This implies that to manage an unsorted database on an actual quantum computer is possible and efficient.
Quantum mechanics can speed up a range of search applications over unsorted data. For example, imagine a phone directory containing N names arranged in completely random order. To find someone's phone number with a probability of 50%, any classical algorithm (whether deterministic or probabilistic) will need to access the database a minimum of 0.5N times. Quantum mechanical systems can be in a superposition of states and simultaneously examine multiple names. By properly adjusting the phases of various operations, successful computations reinforce each other while others interfere randomly. As a result, the desired phone number can be obtained in only O(N) accesses to the database.
IntroductionFor a search problem associated with a unsorted database, the remarkable Grover's quantum algorithm (1) provides a quadratic speedup over its classical counterpart.The search problem can be described as follows: for a given function f , there exists one unknown element in the set {0, 1, ..., N -1} that satisfies f (x)=-1, say x = τ, whereas the other N -1o n e sg i v e f (x)=1.How many times of evaluations of f are required to determine the element τ for f (τ)=-1?Through a conventional algorithm, one needs O(N) trials to achieve this aim.How about the utility of quantum algorithm for the search?For the scenario in the quantum world, the search problem can be rephrased in the quantum mechanical language: for a given unitary operator I τ , that is sometimes called the oracle operator, and a set of state vectors (orthonormal basis): s = {|0 , |1 , ..., |N -1 }, I τ |x = |x for all states in the set except I τ |x = -|x for x = τ.H owma n yqueriesofI τ are required to determine |τ ?By Grover's algorithm, one needs only O( √ N) quantum mechanical steps to find the marked state |τ out.It has been shown that Grover's algorithm is optimal since it needs minimal oracle calls to perform a quantum search (2).The quantum searching process will be briefly reviewed as follows.The first step of Grover's algorithm is to prepare a superposition state of all elements with uniform probability amplitude:Then apply the Grover kernel G = -I η I τ to |s ,w h er eI η is a unitary operator and contains no bias against the marked state.For large N,a f t e ra b o u tm = π √ N/4 repetitions of G operations, the probability to observe |τ is close to one, i.e., G m |s ∼ |τ .( 2 ) Since every single G involves one query of I τ ,onlyO( √ N) searching steps are required for a quantum search task.In what follows, we will first investigate on the general SU(2) formulation for the kernel of Grover's searching operator G.The discussions of quantum searching certainty, robustness, and the analog analogue version of the Grover's algorithm will be given afterwards.
A proposed quantum-computing algorithm would perform a search for an item of information in a database stored in a Hilbert-space memory structure. The algorithm is intended to make it possible to search relatively quickly through a large database under conditions in which available computing resources would otherwise be considered inadequate to perform such a task. The algorithm would apply, more specifically, to a relational database in which information would be stored in a set of N complex orthonormal vectors, each of N dimensions (where N can be exponentially large). Each vector would constitute one row of a unitary matrix, from which one would derive the Hamiltonian operator (and hence the evolutionary operator) of a quantum system. In other words, all the stored information would be mapped onto a unitary operator acting on a quantum state that would represent the item of information to be retrieved. Then one could exploit quantum parallelism: one could pose all search queries simultaneously by performing a quantum measurement on the system. In so doing, one would effectively solve the search problem in one computational step. One could exploit the direct- and inner-product decomposability of the unitary matrix to make the dimensionality of the memory space exponentially large by use of only linear resources. However, inasmuch as the necessary preprocessing (the mapping of the stored information into a Hilbert space) could be exponentially expensive, the proposed algorithm would likely be most beneficial in applications in which the resources available for preprocessing were much greater than those available for searching.
Sorting is a fundamental computational process that facilitates subsequent searching of a database. It can be thought of as factorization of the search process. The location of a desired item in a sorted database can be found by classical queries that inspect one letter of the label at a time. For an unsorted database, no such classical quick search algorithm is available. If the database permits quantum queries, however, then mere digitization is sufficient for efficient search. Sorting becomes redundant with the quantum superposition of states. A quantum algorithm is written down that locates the desired item in an unsorted database a factor of two faster than the best classical algorithm can in a sorted database. This algorithm has close resemblance to the assembly process in DNA replication.