Johannes Gütschow ; Vincent Nesme ; Reinhard F. Werner

The fractal structure of cellular automata on abelian groups
dmtcs:2759 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2010,
DMTCS Proceedings vol. AL, Automata 2010  16th Intl. Workshop on CA and DCS

https://doi.org/10.46298/dmtcs.2759
The fractal structure of cellular automata on abelian groups
Authors: Johannes Gütschow ^{1}; Vincent Nesme ^{1}; Reinhard F. Werner ^{1}
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Johannes Gütschow;Vincent Nesme;Reinhard F. Werner
1 Institut für Theoretische Physik [Hannover]
It is a wellknown fact that the spacetime diagrams of some cellular automata have a fractal structure: for instance Pascal's triangle modulo $2$ generates a Sierpinski triangle. Explaining the fractal structure of the spacetime diagrams of cellular automata is a much explored topic, but virtually all of the results revolve around a special class of automata, whose main features include irreversibility, an alphabet with a ring structure and a rule respecting this structure, and a property known as being (weakly) $p$Fermat. The class of automata that we study in this article fulfills none of these properties. Their cell structure is weaker and they are far from being $p$Fermat, even weakly. However, they do produce fractal spacetime diagrams, and we will explain why and how. These automata emerge naturally from the field of quantum cellular automata, as they include the classical equivalent of the Clifford quantum cellular automata, which have been studied by the quantum community for several reasons. They are a basic building block of a universal model of quantum computation, and they can be used to generate highly entangled states, which are a primary resource for measurementbased models of quantum computing.