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For the unsorted database quantum search with the unknown fraction $\lambda$of target items, there are mainly two kinds of methods, i.e., fixed-point ortrail-and-error. (i) In terms of the fixed-point method, Yoder et al. [Phys.Rev. Lett. 113, 210501 (2014)] claimed that the quadratic speedup overclassical algorithms has been achieved. However, in this paper, we point outthat this is not the case, because the query complexity of Yoder's algorithm isactually in $O(1/\sqrt{\lambda_0})$ rather than $O(1/\sqrt{\lambda})$, where$\lambda_0$ is a known lower bound of $\lambda$. (ii) In terms of thetrail-and-error method, currently the algorithm without randomness has to takemore than 1 times queries or iterations than the algorithm with randomlyselected parameters. For the above problems, we provide the first hybridquantum search algorithm based on the fixed-point and trail-and-error methods,where the matched multiphase Grover operations are trialed multiple times andthe number of iterations increases exponentially along with the number oftrials. The upper bound of expected queries as well as the optimal parametersare derived. Compared with Yoder's algorithm, the query complexity of ouralgorithm indeed achieves the optimal scaling in $\lambda$ for quantum search,which reconfirms the practicality of the fixed-point method. In addition, ouralgorithm also does not contain randomness, and compared with the existingdeterministic algorithm, the query complexity can be reduced by about 1/3. Ourwork provides an new idea for the research on fixed-point and trial-and-errorquantum search.

Quantum search for unknown number of target items hybridizing the fixed-point method with the trail-and-error method