  In the present paper we study nonlinear dynamics of quantum quadratic
operators (q.q.o) acting on the algebra of $2\times 2$ matrices $\bm_2(\bc)$.
First, we describe q.q.o. with Haar state as well as quadratic operators with
the Kadison-Schwarz property. By means of such a description we provide an
example of q.q.o. which is not the Kadision-Schwarz operator. Then we study
stability of dynamics of q.q.o.
