Robust Quantum Search with Uncertain Number of Target States.

Yuanye Zhu,Shijie Wei,Bao Yan,Zeguo Wang

doi:10.3390/e23121649

Yuanye Zhu, Shijie Wei + Show 2 more

Open Access

https://doi.org/10.3390/e23121649

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Abstract

The quantum search algorithm is one of the milestones of quantum algorithms. Compared with classical algorithms, it shows quadratic speed-up when searching marked states in an unsorted database. However, the success rates of quantum search algorithms are sensitive to the number of marked states. In this paper, we study the relation between the success rate and the number of iterations in a quantum search algorithm of given , where M is the number of marked state and N is the number of items in the dataset. We develop a robust quantum search algorithm based on Grover–Long algorithm with some uncertainty in the number of marked states. The proposed algorithm has the same query complexity as the Grover’s algorithm, and shows high tolerance of the uncertainty in the ratio . In particular, for a database with an uncertainty in the ratio , our algorithm will find the target states with a success rate no less than .

Highlights

State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, State Key Laboratory of Mathematical Engineering and Advanced Computing, Zhengzhou 450001, China
We develop a robust quantum search algorithm based on Grover–Long algorithm with some uncertainty in the number of marked states
We develop a robust quantum search algorithm, based on the Grover–Long algorithm, which overcomes the problem of not knowing the exact ratio M/N in advance

Summary

Overview of Grover–Long Algorithm

Extract M marked items from an unstructured database with N items by querying O N/M times. After J + 1 steps of iterations, one can obtain the marked states with certainty by measurement. The quantum search algorithm can be described using the SO(3) picture [11,23] instead of SU(2). In this picture, the quantum search operator in Equation (6) corresponds to a rotation in three-dimensional space with the following matrix form. In this picture, the state vector |ψi = ( a + bi )|τ i + (c + di )|τi is represented as. The initial state |ψi i and the marked state |τ i are represented by sin(2β).

Relationship between the Success Rate and Searching Iterations

Robust Quantum Search with Uncertain Number of Targets

Findings

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