  Scrambling is the delocalization of quantum information over a many-body
system and underlies all quantum-chaotic dynamics. We employ discrete quantum
cellular automata as classically simulable toy models of scrambling. We observe
that these automata break ergodicity, i.e. they exhibit quantum scarring. We
also find that the time-scale of scrambling rises with the local Hilbert-space
dimension and obeys a specific combinatorial pattern. We then show that
scarring is mostly suppressed in a semiclassical limit, demonstrating that
semiclassical-chaotic systems are more ergodic.
