  We study discrete quantum dynamics where single evolution step consists of
unitary system transformation followed by decoherence via coupling to an
environment. Often non-Markovian memory effects are attributed to structured
environments whereas here we take a more general approach within discrete
setting. In addition of controlling the structure of the environment, we are
interested in how local unitaries on the open system allow the appearance and
control of memory effects. Our first simple qubit model, where local unitary is
followed by dephasing, illustrates how memory effects arise despite of having
no-structure in the environment the system is coupled with. We then elaborate
this observation by constructing a model for open quantum walk where the
unitary coin and transfer operation is augmented with dephasing of the coin.
The results demonstrate that in the limit of strong dephasing within each
evolution step, the combined coin-position open system always displays memory
effects and their quantity is independent of the structure of the environment.
Our construction makes possible an experimentally realizable open quantum walk
with photons exhibiting non-Markovian features.
