Quantization of Cellular Automata

TL;DR

This paper investigates the conditions under which classical cellular automata can be linearized into quantum cellular automata, revealing that locality constraints enforce reversibility and exploring nonlocality and information transfer limitations.

Contribution

It proves that linearized quantum cellular automata must be reversible due to locality constraints and discusses nonlocality and information transfer in one-dimensional models.

Findings

Quantum cellular automata constructed from classical ones must be reversible.

Locality constraints are highly restrictive in the quantum case.

Nonlocality in one-dimensional models does not necessarily enable nonlocal information transfer.

Abstract

Take a cellular automaton, consider that each configuration is a basis vector in some vector space, and linearize the global evolution function. If lucky, the r esult could actually make sense physically, as a valid quantum evolution; but do es it make sense as a quantum cellular automaton? That is the main question we a ddress in this paper. In every model with discrete time and space, two things ar e required in order to qualify as a cellular automaton: invariance by translatio n and locality. We prove that this locality condition is so restrictive in the q uantum case that every quantum cellular automaton constructed in this way - i. e., by linearization of a classical one - must be reversible. We also discuss some subtleties about the extent of nonlocality that can be encountered in the o ne-dimensional case; we show that, even when the quantized version is non local, still, under some conditions, we may be unable to use this nonlocality to trans mit information nonlocally.