Quantum walk search algorithm for multi-objective searching with iteration auto-controlling on hypercube

Yao-Yao Jiang(姜瑶瑶)^{1}, Peng-Cheng Chu(初鹏程)^{1}, Wen-Bin Zhang(张文彬)^{2}, and Hong-Yang Ma(马鸿洋)^{1,†}

1 School of Science, Qingdao University of Technology, Qingdao 266033, China; 2 School of Information and Control Engineering, Qingdao University of Technology, Qingdao 266033, China

Abstract Shenvi et al. have proposed a quantum algorithm based on quantum walking called Shenvi-Kempe-Whaley (SKW) algorithm, but this search algorithm can only search one target state and use a specific search target state vector. Therefore, when there are more than two target nodes in the search space, the algorithm has certain limitations. Even though a multi-objective SKW search algorithm was proposed later, when the number of target nodes is more than two, the SKW search algorithm cannot be mapped to the same quotient graph. In addition, the calculation of the optimal target state depends on the number of target states m. In previous studies, quantum computing and testing algorithms were used to solve this problem. But these solutions require more Oracle calls and cannot get a high accuracy rate. Therefore, to solve the above problems, we improve the multi-target quantum walk search algorithm, and construct a controllable quantum walk search algorithm under the condition of unknown number of target states. By dividing the Hilbert space into multiple subspaces, the accuracy of the search algorithm is improved from p_{c}=(1/2)-O(1/n) to p_{c}=1-O(1/n). And by adding detection gate phase, the algorithm can stop when the amplitude of the target state becomes the maximum for the first time, and the algorithm can always maintain the optimal number of iterations, so as to reduce the number of unnecessary iterations in the algorithm process and make the number of iterations reach $ t_{\rm f}=(\pi /2)\sqrt{2^{n-2}} $.

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11975132 and 61772295), the Natural Science Foundation of Shandong Province, China (Grant No. ZR2019YQ01), and the Project of Shandong Provincial Higher Educational Science and Technology Program, China (Grant No. J18KZ012). Thanks to Chen Zhenyu, Yan Dandan, and Ma Yulin for their help in the numerical simulation experiment. Special thanks to the scholars who have made important contributions in quantum computing and quantum search algorithm.

Corresponding Authors:
Hong-Yang Ma
E-mail: hongyang_ma@aliyun.com

Cite this article:

Yao-Yao Jiang(姜瑶瑶), Peng-Cheng Chu(初鹏程), Wen-Bin Zhang(张文彬), and Hong-Yang Ma(马鸿洋) Quantum walk search algorithm for multi-objective searching with iteration auto-controlling on hypercube 2022 Chin. Phys. B 31 040307

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