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Chinese Control and Decision Conference (CCDC)

Two improvements in Grover's algorithm

Kai Kang

Published May 1, 2015• 5 citations
Publisher

Abstract

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.

DOI
10.1109/ccdc.2015.7162096

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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.
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.
When the current Grover algorithm is applied to search some objects in an unsorted quantum database,the probability of correct objects usually falls with the increase of the searched objects.The reason for this problem is analyzed in this paper,and an improved measure based on the new phase matching condition is proposed.In the new phase matching condition,the amplitudes of two phase rotations are the same and the directions of two phase rotations are contrary.When the objects are more than one third of the total items,with the new phase matching condition,all objects can be found by at least 25/27 of the probability and by the only one Grover iteration.The validity of the improved measure is proved by experiment.
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 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.