  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 or
trail-and-error. (i) In terms of the fixed-point method, Yoder et al. [Phys.
Rev. Lett. 113, 210501 (2014)] claimed that the quadratic speedup over
classical algorithms has been achieved. However, in this paper, we point out
that this is not the case, because the query complexity of Yoder's algorithm is
actually 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 the
trail-and-error method, currently the algorithm without randomness has to take
more than 1 times queries or iterations than the algorithm with randomly
selected parameters. For the above problems, we provide the first hybrid
quantum search algorithm based on the fixed-point and trail-and-error methods,
where the matched multiphase Grover operations are trialed multiple times and
the number of iterations increases exponentially along with the number of
trials. The upper bound of expected queries as well as the optimal parameters
are derived. Compared with Yoder's algorithm, the query complexity of our
algorithm indeed achieves the optimal scaling in $\lambda$ for quantum search,
which reconfirms the practicality of the fixed-point method. In addition, our
algorithm also does not contain randomness, and compared with the existing
deterministic algorithm, the query complexity can be reduced by about 1/3. Our
work provides an new idea for the research on fixed-point and trial-and-error
quantum search.
