  We study the dynamical behaviour of the quantum cellular automaton of Refs.
[1, 2], which reproduces the Dirac dynamics in the limit of small wavevectors
and masses. We present analytical evaluations along with computer simulations,
showing how the automaton exhibits typical Dirac dynamical features, as the
Zitterbewegung and the scattering behaviour from potential that gives rise to
the so-called Klein paradox. The motivation is to show concretely how pure
processing of quantum information can lead to particle mechanics as an emergent
feature, an issue that has been the focus of solid-state, optical and
atomic-physics quantum simulator.
