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Journal of Systems Engineering and Electronics

Grover quantum searching algorithm based on weighted targets

Panchi Li, Shiyong Li

Published Apr 1, 2008• 11 citations
Publisher

Abstract

The current Grover quantum searching algorithm cannot identify the difference in importance of the search targets when it is applied to an unsorted quantum database, and the probability for each search target is equal. To solve this problem, a Grover searching algorithm based on weighted targets is proposed. First, each target is endowed a weight coefficient according to its importance. Applying these different weight coefficients, the targets are represented as quantum superposition states. Second, the novel Grover searching algorithm based on the quantum superposition of the weighted targets is constructed. Using this algorithm, the probability of getting each target can be approximated to the corresponding weight coefficient, which shows the flexibility of this algorithm. Finally, the validity of the algorithm is proved by a simple searching example.

DOI
10.1016/s1004-4132(08)60093-6

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The current Grover quantum searching algorithm cannot identify the difference in importance of the search targets when it is applied to an unsorted quantum database, and the probability for each search target is equal. To solve this problem, a Grover searching algorithm based on weighted targets is proposed. First, each target is endowed a weight coefficient according to its importance. Applying these different weight coefficients, the targets are represented as quantum superposition states. Second, the novel Grover searching algorithm based on the quantum superposition of the weighted targets is constructed. Using this algorithm, the probability of getting each target can be approximated to the corresponding weight coefficient, which shows the flexibility of this algorithm. Finally, the validity of the algorithm is proved by a simple searching example.
The current Grover quantum searching algorithm can not identify the difference in importance of the marked items when it is applied to an unsorted quantum database, and the probability for each marked item is equal. To solve the problem, a Grover searching algorithm based on the weighted marked items is proposed. Firstly, each marked item is endowed a weight coefficient according to its importance. Applying these different weight coefficients, the marked items are represented as a quantum superposition state. Secondly, the novel Grover searching algorithm based on the quantum superposition of the weighting marked items is constructed. Using this algorithm the probability of getting each marked item can be approximated to the corresponding weight coefficient, which the flexibility of this algorithm is shown. Finally, the validity of the algorithm is validated by a simple searching example.
The current quantum search algorithms do not consider the difference of improtance among the targets.To solve this problem,this paper proposed a quantum search algorithm based on distributed weight target.It analyzed the iteration results by changing the amplitude of initial states in superposition state,and demonstrated the conditions in order to keep the quantum searching algorithm still effective with endowing the weight coefficient to target states.Based on this condition,it constructed a new quantum superposition state containing the weight information of target states,and the new algorithm kept the characters of original Grover algorithm as the same time.The simulation results show that this algorithm can search each target state successfully with the probability of endowed weight coefficient.
As searching targets in an unordered database with Grover's algorithm,difference in marked items is not taken into consideration.If fraction of marked items is greater than 1/4,success probability of search decreases rapidly with increase of marked items.When the fraction of marked items is greater than 1/2,the algorithm is disabled.Aiming at above problems,an improved Grover's algorithm with weighted targets is proposed in which every target is assigned a weight coefficient according to its significance.With these weight coefficients,targets are represented as quantum superpositions which make probability getting target equal to its weight coefficient.An adaptive phase matching method is proposed based on weighted targets.The directions of phase rotations are contrary,and amplitudes of the two phase rotations are determined by inner-product of target quantum superposition and initial state of the system.As the inner-product is greater than(((3-5)/8)~(1/2),) success probability is equal to 1 with two steps of Grover iteration at most.The improved quantum searching algorithm and the new phase matching are verified by an example.
Grover search algorithm is applied to the current number of objectives in quantum unsorted database, the meaning of each goal difference is not taken into account. When the target is more than a quarter of the total items, the probability of the target falls quickly identify and target increases, and when the target is more than half of the total items, the algorithm will fail. To solve these problems, first of all, based on a weighted index improvement is proposed in which each target in accordance with the right given to the importance of the weight coefficient. These different weight coefficients, we give all the target state, which may make the probability of each target equal weights of the quantum superposition. Secondly, an improved phase matching is proposed the phase rotation in both directions is the same, and the inner product of two states by the target amplitude superposition and initial state of the system is determined. When this inner product is 1/2 or more, when using the improved phase matching, the target may be 100% probability and by only one Grover iteration. Finally, the effectiveness of these measures by a simple search instance validation.
Quantum search algorithm proposed by Grover(Grover's algorithm) finds target data with equal probability for all target data. Y. Seo et al. [IEEE Access, vol. 10, pp.16209-16224 (2022)] proposed the quantum search algorithm for weighted solutions to discriminate the importance of target data. In the simple case, we represent the analysis of the algorithm. The low bound of the number of applying Grover operatortttis described using amplitude expression. If we choose an appropriatetttvalue, we provide the maximum ratio of the amplitude of high and low-importance target data. In addition, we arrange the maximum probability of high-importance target data using the bound.
The algorithm proposed by Grover solves the unsorted database search problem with computation complexity of O(2～(1/2)n/m),where the size of database is N=2n and there is m target solutions.However,the successful probability of searching decreases rapidly,when the target solutions m is greater than N/4,And the algorithm is disabled when m is equal to N/2.This paper presents an improved Grover's algorithm,which can find the object with the successful probability at least 98.01% with one step when m is greater than N/4.