提升Grover算法效率：自适应相位匹配优化

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本文主要探讨了Grover算法中的自适应相位匹配技术，该研究发表于2011年的《量子信息科学》期刊（Journal of Quantum Information Science），卷1，页码43-49，DOI:10.4236/jqis.2011.12006。作者Panchi Li 和 Kaoping Song 分别来自中国东北石油大学计算机与信息技术学院和石油工程学院。他们的研究关注的问题是当Grover算法用于搜索无序数据库时，随着标记项（目标项）数量的增加，成功找到目标的概率通常会降低。 Grover算法，作为量子计算中的一种经典搜索算法，其基本原理是利用量子并行性和叠加态的特性来加速搜索过程。然而，传统的Grover迭代过程中，每次迭代的成功概率并不均匀，特别是在标记项比例较高的情况下，效率下降明显。为了克服这个问题，研究者提出了一种自适应相位匹配策略。这个新方法的主要贡献在于，当标记项的比例大于1/5且小于3/8时，通过应用自适应相位匹配，最多只需要两次Grover迭代，就能确保搜索成功。自适应相位匹配的核心思想是根据当前搜索状态动态调整相位因子，以优化量子系统的行为。它不是简单的固定相位调整，而是根据算法的进展和数据库的特性实时进行优化，从而最大化搜索效率。论文作者通过一个具体的搜索示例验证了这种新型相位匹配的有效性，证明了在特定条件下，使用自适应相位匹配可以显著提高Grover算法在实际应用中的性能。关键词包括量子计算、量子搜索、Grover算法、相位匹配以及自适应相位移。这项研究对于理解量子计算的优化策略以及改进现有的量子搜索算法具有重要意义，为处理大规模无序数据提供了新的可能。整体上，这篇论文不仅深入剖析了量子搜索算法的局限性，还为解决这些问题提供了一种创新的解决方案，推动了量子信息科学领域的理论和技术发展。

Journal of Quantum Information Science, 2011, 1, 43-49

doi:10.4236/jqis.2011.12006 Published Online September 2011 (http://www.SciRP.org/journal/jqis)

Adaptive Phase Matching in Grover’s Algorithm

Panchi Li

, Kaoping Song

School of Computer & Information Technology, Northeast Petroleum University, Daqing, China

School of Petroleum Engineering, Northeast Petroleum University, Daqing, China

E-mail:

lipanchi@vip.sina.com

Received June 28, 2011; revised August 16, 2011; accepted August 26, 2011

Abstract

When the Grover’s algorithm is applied to search an unordered database, the successful probability usually

decreases with the increase of marked items. In order to solve this problem, an adaptive phase matching is

proposed. With application of the new phase matching, when the fraction of marked items is greater





35 8

, the successful probability is equal to 1 with at most two Grover iterations. The validity of the

new phase matching is verified by a search example.

Keywords:

Quantum Computing, Quantum Searching, Grover’s Algorithm, Phase Matching, Adaptive Phase

Shifting

1. Introduction

Shor’s prime factoring algorithm and Grover’s quantum

search algorithm are two of the great quantum algorithms

[1,2]. Grover’s search algorithm provides a dramatic

example of the potential speedup offered by quantum

computers [2,3]. The problem addressed by Grover’s

algorithm can be viewed as trying to find a marked ele-

ment in an unsorted database of size N. To solve this

problem, a classical computer would need, on average,

2N database queries and N queries in the worst case.

Using Grover’s algorithm, a quantum computer can ac-

complish the same task using merely





queries.

The importance of Grover’s result stems from the fact

that it proves the enhanced power of quantum computers

compared to classical ones for a whole class of oracle-

based problems, for which the bound on the efficiency of

classical algorithms is known. At present, Grover’s quan-

tum search algorithm has been greatly noticed and has

become a challenging research field. However, the

Grover’s algorithm also has some limitations. When the

fraction of marked items is greater than a quarter of the

total items in the database, the success probability will

rapidly decrease, and when the fraction of marked items

is greater than half of the total items in the database, the

algorithm will be disabled.

Up to now, many efforts in improving Grover’s origi-

nal algorithm have been done. Boyer et al. have given

analytical expressions for the amplitude of the states in

Grover’s search algorithm and tight bounds on the algo-

rithm [4]. Zalka has improved these tight bounds and

showed that Grover’s algorithm is optimal [5]. Biron et

al. generalized Grover’s algorithm to an arbitrarily dis-

tributed initial state [6]. Pati recast the algorithm in geo-

metric language and studied the bounds on the algorithm

[7]. Ozhigov showed that quantum search can be further

speeded up by a factor of

2 by parallelism [8]. Gin-

grich et al. also generalized Grover’s algorithm with par-

allelism with improvement [9]. The Grover’s original

algorithm consists of inversion of the amplitude in the

desired state and inversion-about-average operation [2].

In [10], Grover presented a general algorithm:



UIU





 , where U is any unitary operation, U



is the

adjoint of U,

2II



 , 2II



 ,



is an initial state and



is a desired state. When





 , where H is the Walsh-Hadamard trans-

formation, and



 , the general algorithm becomes

the original algorithm. Long extended Grover’s algo-

rithm [11]. In Long’s algorithm,



and



are ex-

pressed as





1II







 and













 , respectively. When π





, Long’s algo-

mes Grover’s original algorithm. Li et al. pro-

posed that U in Long’s algorithm can be replaced by any

unitary operation V [12,13]. Biham generalized the

Grover’s algorithm to deal with an arbitrary pure initial

state and an arbitrary mixed initial state [14,15]. In [16],

Grover presented the new algorithm by replacing the

selective inversions by selective phase shifts of

rithm beco

π 3 .

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