  A generalized set of Clifford cellular automata, which includes all Clifford
cellular automata, result from the quantization of a lattice system where on
each site of the lattice one has a $2k$-dimensional torus phase space. The
dynamics is a linear map in the torus variables and it is also local: the
evolution depends only on variables in some region around the original lattice
site. Moreover it preserves the symplectic structure. These are classified by
$2k\times 2k$ matrices with entries in Laurent polynomials with integer
coefficients in a set of additional formal variables. These can lead to fractal
behavior in the evolution of the generators of the quantum algebra. Fractal
behavior leads to non-trivial Lyapunov exponents of the original linear
dynamical system. The proof uses Fourier analysis on the characteristic
polynomial of these matrices.
