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.

4 pages, plus supplementary material. v2: references added

References in corpus (5)

• Operator Spreading in Random Unitary Circuits
• Bounds on chaos from the eigenstate thermalization hypothesis
• On the structure of Clifford quantum cellular automata
• A toy model for time evolving QFT on a lattice with controllable chaos
• Exotic equilibration dynamics on a 1-D quantum CNOT gate lattice