  In the present paper we study description of Kadison-Schwarz type quantum
quadratic operators acting from $\bm_2(\mathbb{C})$ into
$\bm_2(\mathbb{C})\o\bm_2(\mathbb{C})$. Note that such kind of operator is a
generalization of quantum convolution. By means of such a description we
provide an example of q.q.o. which is not a Kadision-Schwartz operator.
Moreover, we study dynamics of an associated nonlinear (i.e. quadratic)
operators acting on the state space of $\bm_2(\mathbb{C})$.
