  We show that the Dirac quantum cellular automaton [Ann. Phys. 354 (2015) 244]
shares many properties in common with the discrete-time quantum walk. These
similarities can be exploited to study the automaton as a unitary process that
takes place at regular time steps on a one-dimensional lattice, in the spirit
of general quantum cellular automata. In this way, it becomes an alternative to
the quantum walk, with a dispersion relation that can be controlled by a
parameter, which plays a similar role to the coin angle in the quantum walk.
The Dirac Hamiltonian is recovered under a suitable limit. We provide two
independent analytical approximations to the long term probability
distribution. It is shown that, starting from localized conditions, the
asymptotic value of the entropy of entanglement between internal and motional
degrees of freedom overcomes the known limit that is approached by the quantum
walk for the same initial conditions, and are similar to the ones achieved by
highly localized states of the Dirac equation.

Quantum Physics