  A class of models with a dynamics of generalized quantum cat maps on a
product of quantum tori is described. These tori are defined by an algebra of
clock-shift matrices of dimension $N$. The dynamics is such that the Lyapunov
exponents can be computed analytically at large $N$. Some of these systems can
be thought of as a toy model for quantum fields on a lattice under a time
evolution with nearest neighbor interactions, resembling a quantum version of a
cellular automaton. The dynamics of entangling is studied for initial product
states. Some of these entangle at rates determined by Lyapunov exponents of the
system at large $N$ when the initial states are gaussian. For other classes of
states, entanglement between two regions can be computed analytically: it is
found that entanglement rates are controlled by $\log(N)$. Some of these setups
can be realized on quantum computers with CNOT quantum gates. This is analyzed
in detail where we find that the dynamics has a self-similar behavior and
various peculiar behaviors. This dynamics can be interpreted in a particular
basis as a machine that broadcasts classical messages in one direction and that
produces over time a generalized GHZ state with the receiving region once we
consider superpositions of such messages. With the appropriate choice of
quantum vacuum on the receiving end of the system one can stop the message from
leaving the broadcast area.
