Non-trivial Lyapunov spectrum from fractal quantum cellular automata

TL;DR

This paper explores how certain quantum cellular automata exhibit fractal dynamics leading to complex Lyapunov spectra, using Fourier analysis to classify their behavior and analyze stability.

Contribution

It introduces a classification of Clifford cellular automata derived from lattice quantization, revealing fractal evolution and non-trivial Lyapunov exponents.

Findings

Fractal behavior in quantum automata evolution.

Classification of automata via symplectic matrices with Laurent polynomial entries.

Identification of non-trivial Lyapunov spectra in these systems.

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.