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It is generally believed that Grover search algorithm (GSA) with quantum noise may quickly lose its quadratic speedup over its classical case. In this paper, we partly agree with that by our new findings as follows. First, we investigate different typical diagonalizable noises represented by Bloch vectors, and the results demonstrate that the success probability decreases exponentially to 1/2 and oscillates around 1/2 with the increase of the number of iterations. Second, for some types of noises, such as bit flip and bit-phase flip noises, can improve the performance of GSA for certain parts of the search process. Third, we calculate and analyze the noise threshold of the bit-phase flip noise for the requested success probability and the result shows that GSA with noise within the threshold still outperforms its classical counterpart. According to the above results, some interesting works in the noisy intermediate-scale quantum (NISQ) computing are suggested, such as verifying the correctness of quantum algorithms even with noises and machine learning applications.

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