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AN ASSOCIATIVE MODEL BASED ON QUANTUM SEARCH ALGORITHMS

Hiromi Miyajima, Noritaka Shigei Masataka Fujisaki

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

In this paper, an algorithm of quantum associative memory is proposed using quantum search algorithms. In the associative memory, the problem is to find the data or the nearest data in the meaning of hamming distance from designated data. If the search data in memorized dataset is contained, the problem becomes data search one. In associative memory, it is difficult to find the nearest data in hamming distance. For searching any item in an unsorted database with N items, a classical takes O(N) steps but Grover’s quantum searching algorithm takes only O( p N) steps on perfect graph. By generalizing the idea, Ventura has proposed the associative memory model, but it does not solve the problem of finding the nearest data. On the other hand, quantum search algorithm with O( p N) steps on hypercube is also proposed. Therefore, the idea in this paper is formalizing the problem as quantum search algorithm in hypercube but not in the conventional perfect graph. First, we perform quantum search algorithm by considering quantum walk on hypercube. Further, it is observed search data or the nearest data by using the conventional algorithm with high probability.

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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.

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This article reviews recent progress in quantum database search algorithms. The subject is presented in a self-contained and pedagogical way. The problem of searching a large database (a Hilbert space) for a target item is performed by the famous Grover algorithm which locates the target item with high probability and a quadratic speed-up compared with the corresponding classical algorithm. If the database is partitioned into blocks and one is searching for the block containing the target item instead of the target item itself, then the problem is referred to as partial search. Partial search trades accuracy for speed and the most efficient version is the Grover–Radhakrishnan–Korepin (GRK) algorithm. The target block can be further partitioned into sub-blocks so that GRK's can be performed in a sequence called a hierarchy. We study the Grover search and GRK partial search in detail and prove that a GRK hierarchy is less efficient than a direct GRK partial search. Both the Grover search and the GRK partial search can be generalized to the case with several target items (or target blocks for a GRK). The GRK partial search algorithm can also be represented in terms of group theory.

International Journal of Modern Physics B•Dec 20, 2009•7 citations
In this paper, an algorithm of quantum associative memory is proposed using quantum search algorithms. In the associative memory, the problem is to find the data or the nearest data in the meaning of hamming distance from designated data. If the search data in memorized dataset is contained, the problem becomes data search one. In associative memory, it is difficult to find the nearest data in hamming distance. For searching any item in an unsorted database with N items, a classical takes O(N) steps but Grover’s quantum searching algorithm takes only O( p N) steps on perfect graph. By generalizing the idea, Ventura has proposed the associative memory model, but it does not solve the problem of finding the nearest data. On the other hand, quantum search algorithm with O( p N) steps on hypercube is also proposed. Therefore, the idea in this paper is formalizing the problem as quantum search algorithm in hypercube but not in the conventional perfect graph. First, we perform quantum search algorithm by considering quantum walk on hypercube. Further, it is observed search data or the nearest data by using the conventional algorithm with high probability.
For searching any item in an unsorted database with N items, a classical computer takes O(N) steps but Grover’s quantum searching algorithm takes only O( √ N) steps. However, it is also known that Grover’s algorithm is effective only in the case where the initial amplitude distribution of dataset is uniform, but is not always effective in the nonuniform case. In this paper, we propose some quantum search algorithms. First, we propose an algorithm in analog time based on quantum walk by solving the schrodinger equation. The proposed algorithm shows best performance in optimum time. Next, we will apply the result to Grover search algorithm. It is shown that the proposed algorithm shows better performance than the conventional one. Further, we propose the improved algorithm by introducing the idea of the phase rotation. The algorithm shows best performance compared with the conventional ones.
For searching an unsorted database with N items, a classical computer requires O(N) steps but Grover's quantum searching algorithm requires only O(¿N) steps. However, it is also known that Grover's algorithm is effective in the case where the initial amplitude distribution of dataset is uniform, but is not always effective in the non-uniform case. In this paper, we will consider the influence of initial amplitude distribution of dataset for Grover's and Ventura's algorithms. Further, in order to improve the performance of quantum searching, we propose a new algorithm and analyze its dynamics.
As a new quantum computing model, quantum walk has been widely studied in recent years. It consists of discrete time quantum walk and continuous time quantum walk. Discrete time quantum walk includes coin quantum walk and scattering quantum walk. Meanwhile as a quantum search algorithm, Grover algorithm can search an unsorted database in time complexity of O(√N) . Recent years quantum walk has been used to solve search problems and some of them have been proved to be as efficient as Grover algorithm. Making full use of the novel properties of quantum walk, the quantum walk search algorithms on the 2D grid and hypercube have been proposed. Inspired by phase matching condition of Grover algorithm, we propose a new quantum walk search algorithm which is based on coin quantum walk. Firstly we give the graph to the quantum walk, and then describe the algorithm in detail. Algorithm uses different coin operators and shift operators for two different cases and draws the corresponding iteration operators. Then we prove that iteration operators used in the algorithm are unitary operators. Then we analyze the time complexity and probability of success of the algorithm. Analysis indicates that the time complexity of the algorithm is the same as that of Grover algorithm, however when the targets to be searched are more than 1/3 of the total targets, the algorithm probability of success is greater than that of Grover algorithm. Finally we give the circuit implementation of the algorithm.
For searching any item in an unsorted database with N items, Grover's quantum searching algorithm takes only O(√N) steps, which is much faster than O(N) steps required for a classical computer. However, it is known that Grover's algorithm is effective only in the case where the initial amplitude distribution of dataset is uniform. On the other hand, Ventura has also proposed a quantum searching algorithm. The algorithm is effective only in the special case for the initial amplitude distribution. Therefore, we have proposed an effective quantum searching algorithm in another case in the previous paper. In this paper, we generalize the previous algorithm and propose several effective algorithms in several cases where initial amplitude distributions of dataset are non-uniform. Further, in order to show the effectiveness of the algorithms, we analyze their dynamics.
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.
This article reviews recent progress in quantum database search algorithms. The subject is presented in a self-contained and pedagogical way. The problem of searching a large database (a Hilbert space) for a target item is performed by the famous Grover algorithm which locates the target item with high probability and a quadratic speed-up compared with the corresponding classical algorithm. If the database is partitioned into blocks and one is searching for the block containing the target item instead of the target item itself, then the problem is referred to as partial search. Partial search trades accuracy for speed and the most efficient version is the Grover–Radhakrishnan–Korepin (GRK) algorithm. The target block can be further partitioned into sub-blocks so that GRK's can be performed in a sequence called a hierarchy. We study the Grover search and GRK partial search in detail and prove that a GRK hierarchy is less efficient than a direct GRK partial search. Both the Grover search and the GRK partial search can be generalized to the case with several target items (or target blocks for a GRK). The GRK partial search algorithm can also be represented in terms of group theory.