  The search operation for a marked state by means of Grover's quantum
searching algorithm is shown to be an element of group SU(2) which acts on a
2-dimensional space spanned by the marked state and the unmarked collective
state. Based on this underlying structure, those exact bounds of the steps in
various quantum search algorithms are obtained in a quite concise way. This
reformulation of the quantum searching algorithm also enables a detailed
analysis of the decoherence effects caused by its coupling with an environment.
It turns out that the environment will not only make the quantum search invalid
in case of complete decoherence, where the probability of finding the marked
state is unchanged, but also it may make the quantum search algorithm worse
than expected: It will decrease this probability when the environment shows its
quantum feature.
