  We analyze the asymptotic dynamics of quantum systems resulting from large
numbers of iterations of random unitary operations. Although, in general, these
quantum operations cannot be diagonalized it is shown that their resulting
asymptotic dynamics is described by a diagonalizable superoperator. We prove
that this asymptotic dynamics takes place in a typically low dimensional
attractor space which is independent of the probability distribution of the
unitary operations applied. This vector space is spanned by all eigenvectors of
the unitary operations involved which are associated with eigenvalues of unit
modulus. Implications for possible asymptotic dynamics of iterated random
unitary operations are presented and exemplified in an example involving random
controlled-not operations acting on two qubits.
