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Chinese Physics Letters

A Four-Phase Improvement of Grover’s Algorithm

Bo-wen Ma, Wan‐Su Bao, Tan Li, Feng‐Guang Li, Shuo Zhang, Xiang-Qun Fu

Published Jul 1, 2017• 3 citations
Publisher

Abstract

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%.

DOI
10.1088/0256-307x/34/7/070305

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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 applying Grover's algorithm to an unordered database, the probability of obtaining correct results usually decreases as the quantity of target increases. To amend the limitation, numbers of improved schemes are proposed. In this paper, we focus on four improved schemes from phases, and find that they are just differed by a global phase. Based on this conclusion, the extensive researches on one scheme can be easily generated to other three schemes, and some examples are presented to indicate the correctness.
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 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.
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.
When the Grover’s algorithm is applied to search an unordered database, the probability of success usually decreases with the increase of marked items. To address this phenomenon, a fixed-phase quantum search algorithm with more flexible behavior is proposed. In proposed algorithm, the phase shifts can be fixed at the different values to meet the needs of different practical problems. If research requires a relatively rapid speed, the value of the phase shifts should be appropriately increased, if search requires a higher success probability, the value of the phase shifts should be appropriately decreased. When the phase shifts are fixed at , the success probability of at least 99.38% can be obtained in iterations.