基于相位匹配的量子行走搜索算法及电路实现

Quantum walk search algorithm based on phase matching and circuit cmplementation

量子行走是经典随机行走在量子力学框架下的对应,理论上可以用来解决一类无序数据库的搜索问题.因为携带信息的量子态的扩散速度与经典相比有二次方式的增长,所以量子行走优于经典随机行走,量子行走的特性值得加以利用.量子行走作为一种新发现的物理现象的数学描述,引发了一种新的思维方式,孕育了一种新的理论计算模型.最新研究表明,量子行走本身也是一种通用计算模型,可被视为设计量子算法的高级工具,因此受到部分计算机理论科学领域学者的关注和研究.对于多数问题求解方案的量子算法的设计,理论上可以只在量子行走模型下进行考虑.基于Grover算法的相位匹配条件,本文提出了一个新的基于量子行走的搜索算法.理论演算表明:一般情况下本算法的时间复杂度与Grover算法相同,但是当搜索的目标数目多于总数的1/3时,本算法搜索成功的概率要大于Grover算法.本文不但利用Grover算法中相位匹配条件构造了一个新的量子行走搜索算法,而且在本研究室原有的量子电路设计研究成果的基础上给出了该算法的量子电路表述.

As a new quantum computing model, quantum walk has been widely studied in recent years. It consists of discrete time quantum walk and continuous time quantum walk. Discrete time quantum walk includes coin quantum walk and scattering quantum walk. Meanwhile as a quantum search algorithm, Grover algorithm can search an unsorted databasein time complexity of O（√N）. Recent years quantum walk has been used to solve search problems and some of them have been proved to be as efficient as Grover algorithm. Making full use of the novel properties of quantum walk, the quantum walk search algorithms on the 2D grid and hypercube have been proposed. Inspired by phase matching condition of Grover algorithm, we propose a new quantum walk search algorithm which is based on coin quantum walk. Firstly we give the graph to the quantum walk, and then describe the algorithm in detail. Algorithm uses different coin operators and shift operators for two different cases and draws the corresponding iteration operators. Then we prove that iteration operators used in the algorithm are unitary operators. Then we analyze the time complexity and probability of success of the algorithm. Analysis indicates that the time complexity of the algorithm is the same as that of Grover algorithm,however when the targets to be searched are more than 1/3 of the total targets, the algorithm probability of success is greater than that of Grover algorithm. Finally we give the circuit implementation of the algorithm.