On Quantum Cellular Automata

TL;DR

This paper proposes a general scheme for quantizing classical cellular automata, including irreversible and reversible types, using Hilbert space vectors, and discusses related quantum models like lattice gases and quantum walks.

Contribution

It introduces a novel approach to define quantum cellular automata with a history scheme, extending previous algebraic models to more general classical CA.

Findings

A new scheme for quantum CA based on local transition rules

Comparison between Hilbert space approach and C*-algebra methods

Discussion of quantum lattice gases and quantum walks

Abstract

In recent work [quant-ph/0405174] by Schumacher and Werner was discussed an abstract algebraic approach to a model of reversible quantum cellular automata (CA) on a lattice. It was used special model of CA based on partitioning scheme and so there is a question about quantum CA derived from more general, standard model of classical CA. In present work is considered an approach to definition of a scheme with "history", valid for quantization both irreversible and reversible classical CA directly using local transition rules. It is used language of vectors in Hilbert spaces instead of C*-algebras, but results may be compared in some cases. Finally, the quantum lattice gases, quantum walk and "bots" are also discussed briefly.