  One can think of some physical evolutions as being the emergent-effective
result of a microscopic discrete model. Inspired by classical coarse-graining
procedures, we provide a simple procedure to coarse-grain color-blind quantum
cellular automata that follow Goldilocks rules. The procedure consists in (i)
space-time grouping the quantum cellular automaton (QCA) in cells of size $N$;
(ii) projecting the states of a cell onto its borders, connecting them with the
fine dynamics; (iii) describing the overall dynamics by the border states, that
we call signals; and (iv) constructing the coarse-grained dynamics for
different sizes $N$ of the cells. A byproduct of this simple toy-model is a
general discrete analog of the Stokes law. Moreover we prove that in the
spacetime limit, the automaton converges to a Dirac free Hamiltonian. The QCA
we introduce here can be implemented by present-day quantum platforms, such as
Rydberg arrays, trapped ions, and superconducting qbits. We hope our study can
pave the way to a richer understanding of those systems with limited
resolution.
