Abstract: In recent years a great effort has been put into elucidating the role of quantum mechanics in macroscopical thermodynamical phenomena. One of key topics is the thermalization of physical systems, so common in an everyday life. An important ingredient of the thermalization is the equilibration of the system, i.e. its relaxation towards a stationary final state. We have studied the equilibration of a manybody quantum-system model, the quantum network, governed by two kinds of mutual interactions, CNOT interaction and partial swap interaction. In the presentation we summarize the asymptotic state of the network and conditions under which the network equilibrates.