  We investigate the asymptotic dynamics of quantum networks under repeated
applications of random unitary operations. It is shown that in the asymptotic
limit of large numbers of iterations this dynamics is generally governed by a
typically low dimensional attractor space. This space is determined completely
by the unitary operations involved and it is independent of the probabilities
with which these unitary operations are applied. Based on this general feature
analytical results are presented for the asymptotic dynamics of arbitrarily
large cyclic qubit networks whose nodes are coupled by randomly applied
controlled-NOT operations.
