  We analyze random unitary evolution of the qubit within memory kernel
approach. We provide sufficient conditions which guarantee that the
corresponding memory kernel generates physically legitimate quantum evolution.
Interestingly, we are able to recover several well known examples and generate
new classes of nontrivial qubit evolution. Surprisingly, it turns out that
quantum evolution with memory kernel generated by our approach gives rise to
vanishing non-Markovianity measure based on the distinguishability of quantum
states.
