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Chinese Control Conference (CCC)

Phase matching in grover's algorithm

Xin Li, Kaoping Song, Ning Sun, Chunli Zhao

Published Jul 26, 2013• 1 citation
OpenAlex

Abstract

When the Grover's algorithm is applied to search an unordered database, the probability of getting correct results usually decreases with the increase of marked items. The reason for this phenomenon is analyzed in this Letter, the Grover iteration is studied, and a new phase matching is proposed. With application of the new phase matching, when the fraction of marked items is greater than 1/3, the probability of getting correct results is greater than 25/27 with only one Grover iteration. The validity of the new phase matching is verified by a search example.

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When the Grover's algorithm is applied to search an unordered database, the probability of getting correct results usually decreases with the increase of marked items. The reason for this phenomenon is analyzed in this Letter, the Grover iteration is studied, and a new phase matching is proposed. With application of the new phase matching, when the fraction of marked items is greater than 1/3, the probability of getting correct results is greater than 25/27 with only one Grover iteration. The validity of the new phase matching is verified by a search example.
When the Grover's algorithm is applied to search an unordered database, the success probability usually decreases with the increase of marked items. The reason for this phenomenon was analyzed, the Grover iteration was studied, and a new adaptive phase rotation was proposed. With application of the new phase rotation, when the fraction of marked items is a rational number in range (3-5~(1/2)) / 8to 1/4, the success probability is equal to 1 after two Grover iterations; and when the fraction of marked items is greater than1/4, the success probability is equal to 1 after the only one Grover iteration. The effectiveness of the new phase rotation is verified by a search example.
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 applying Grover’s algorithm to an unordered database, the probability of obtaining correct results usually decreases as the quantity of target increases. A four-phase improvement of Grover’s algorithm is proposed to fix the deficiency, and the unitary and the phase-matching condition are also proposed. With this improved scheme, when the proportion of target is over 1/3, the probability of obtaining correct results is greater than 97.82% with only one iteration using two phases. When the computational complexity is , the algorithm can succeed with a probability no less than 99.63%.
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.
Phase matching has been studied for the Grover algorithm as a way of enhancing the efficiency of the quantum search. Recently Li and Li found that a particular form of phase matching yields, with a single Grover operation, a success probability greater than 25/2725/2725/27 for finding the equal-amplitude superposition of marked states when the fraction of the marked states stored in a database state is greater than 1/31/31/3. Although this single operation eliminates the oscillations of the success probability that occur with multiple Grover operations, the latter oscillations reappear with multiple iterations of Li and Li's phase matching. In this paper we introduce a multiphase matching subject to a certain matching rule by which we can obtain a multiple Grover operation that with only a few iterations yields a success probability that is almost constant and unity over a wide range of the fraction of marked items. As an example we show that a multiphase operation with six iterations yields a success probability between 99.8 and 100 100\text{ }%100  for a fraction of marked states of 1/101/101/10 or larger.