Feedback search algorithm for multi-particle quantum walks over a ring based on permutation groups

Abstract

In quantum computing science, much attention has been paid to how to construct quantum search algorithms better, moreover, searching for new search algorithms based on quantum walk is still attracting scholars' continuous in-depth research and exploration. In this paper, a multi-particle quantum walk search algorithm based on permutation group is proposed by considering many aspects, such as reducing time consumption and increasing the accuracy and controllability of algorithm search. Firstly, the permutation group can be regarded as a closed loop in space, and the permutation set is defined. The data set of data points is mapped to the defined permutation set by isomorphism mapping, which makes the element data points in the permutation set form a one-to-one correspondence. Secondly, according to the given initial state and coin operator, the target data search is carried out on the ring by using the quantum walk of multiple particles in the search space of the data point set and the permutation set. Finally, the target data is found according to the function <inline-formula><tex-math id="M3">\begin{document}$\varPhi(w)=1 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211000_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211000_M3.png"/></alternatives></inline-formula>, and the value is stored by the quantum state, which is used to form the feedback control of the search algorithm. At the same time, the direction of quantum walking on the ring is controlled by controlling the coin operator, thus increasing the operability and accuracy of the search. In this paper, the quantum walk of multiple particles is used to search, and the analysis shows that the particle number parameter <i>j</i> is negatively correlated with the time complexity, but not negatively linear. The proposed quantum walk search algorithm conforms to the zero condition and the lower bound condition, and is not affected by the variable parameter <i>j</i>. Through numerical analysis, it is found that the time complexity of the quantum walk search algorithm is equivalent to <inline-formula><tex-math id="M4">\begin{document}$O(\sqrt[3]{N})$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211000_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211000_M4.png"/></alternatives></inline-formula>, which improves the search efficiency compared with the Grover search algorithm.