  We propose a new finding $k$-minima algorithm and prove that its query
complexity is $\mathcal{O}(\sqrt{kN})$, where $N$ is the number of data
indices. Though the complexity is equivalent to that of an existing method, the
proposed is simpler. The main idea of the proposed algorithm is to search a
good threshold that is near the $k$-th smallest data. Then, by using the
generalization of amplitude amplification, all $k$ data are found out of order
and the query complexity is $\mathcal{O}(\sqrt{kN})$. This generalization of
amplitude amplification is also not well discussed and we briefly prove the
query complexity. Our algorithm can be directly adapted to distance-related
problems like $k$-nearest neighbor search and clustering and classification.
There are few quantum algorithms that return multiple answers and they are not
well discussed.
