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arXiv:1801.07369

Research on phases of Grover-type algorithms

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

Published Jan 23, 2018• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1801.07369

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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. 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 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 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 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.
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.
We briefly introduced some of our recent work related to the phase matching condition in quantum searching algorithms and the improved Grover algorithm. When one replaces the two phase inversions in the Grover algorithm with arbitrary phase rotations, the modified algorithm usually fails in searching the marked state unless a phase matching condition is satisfied between the two phases. The Grover algorithm is not 100% in success rate, an improved Grover algorithm with zero-failure rate is given by replacing the phase inversions with angles that depends on the size of the database. Other aspects of the Grover algorithm such as the SO(3) picture of quantum searching, the dominant gate imperfections in the Grover algorithm are also mentioned.