Non-trivial Lyapunov spectrum from fractal quantum cellular automata

Abstract

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.