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International Conference on Innovative Computing Technology

Sorting N elements using quantum entanglement sets

Ammar Odeh, Khaled M. Elleithy, Marwah Mohammad Almasri, Abrar Mohammed Alajlan

Published Aug 1, 2013• 7 citations
Publisher

Abstract

To search for an element in an unsorted database, it requires O(N) operations using classical computers, where it takes only O(√ N) operations in quantum systems. In this paper, we provide an in-depth look at the existing quantum searching algorithms and emphasize the quantum entanglement feature to propose a new technique for sorting N elements and thus improve the sorting process.

DOI
10.1109/intech.2013.6653693

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To search for an element in an unsorted database, it requires O(N) operations using classical computers, where it takes only O(√ N) operations in quantum systems. In this paper, we provide an in-depth look at the existing quantum searching algorithms and emphasize the quantum entanglement feature to propose a new technique for sorting N elements and thus improve the sorting process.
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.
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.
Given two unsorted lists each of length N that have a single common entry, a quantum computer can find that matching element with a work factor of O(N3/4log⁡N)O(N^{3/4}\log N)O(N3/4logN) (measured in quantum memory accesses and accesses to each list). The amount of quantum memory required is O(N1/2)O(N^{1/2})O(N1/2). The quantum algorithm that accomplishes this consists of an inner Grover search combined with a partial sort all sitting inside of an outer Grover search.
Abstract. Quantum computation, in particular Grover’s algorithm, has aroused a great deal of interest since it allows for a quadratic speedup to be obtained in search procedures. Classical search procedures for an N element database require at most O(N) time complexity. Grover’s algorithm is able to find a solution with high prob-ability in O( N) time through an amplitude amplification scheme. In this work we draw elements from both classical and quantum computation to develop an alternative search proposal based on quantum entanglement detection schemes. In 2002, Horodecki and Ekert proposed an efficient method for direct detection of quantum entanglement. Our proposition to quantum search combines quantum entanglement detection along-side entanglement inducing operators. Grover’s quantum search relies on measuring a quantum superposition after having applied a unitary evolution. We deviate from the standard method by focusing on fine-tuning a unitary operator in order to infer the solution with certainty. Our proposal sacrifices space for speed and depends on the mathematical properties of linear positive maps Λ which have not been operationally characterized. Whether such a Λ can be easily determined remains an open question.
Quantum entanglement, a bizarre and fundamental property of quantum mechanics, presents a potentially revolutionary approach to computation and search. This paper explores the development of a novel quantum search algorithm predicated on entanglement, aiming to surpass the limitations of traditional methods by offering enhanced efficiency and potential for tackling complex data sets. We detail the core mechanism of this algorithm, emphasizing its reliance on the unique correlations inherent in entangled states, and present a preliminary analysis of its performance characteristics. The investigation will focus on demonstrating the algorithm's capability to efficiently locate specific data points within a large dataset, highlighting the conceptual advantages over existing search techniques. The core claim is that this quantum search algorithm achieves significantly improved performance, particularly in scenarios with high dimensionality or complex search spaces. We conclude with a discussion of potential future directions and challenges in implementing this algorithm.