Quantum-inspired identification of complex cellular automata

TL;DR

This paper introduces a quantum-inspired method to classify elementary cellular automata based on how quickly they generate complex structures, using quantum statistical memory as a measure.

Contribution

It applies quantum stochastic modelling to cellular automata, providing a new quantitative framework to identify and rank complex behaviour in ECA rules.

Findings

Quantum statistical memory distinguishes simple and complex ECA.

Growth rate of quantum memory correlates with automata complexity.

Provides a spectrum for ranking ECA complexity.

Abstract

Elementary cellular automata (ECA) present iconic examples of complex systems. Though described only by one-dimensional strings of binary cells evolving according to nearest-neighbour update rules, certain ECA rules manifest complex dynamics capable of universal computation. Yet, the classification of precisely which rules exhibit complex behaviour remains a significant challenge. Here we approach this question using tools from quantum stochastic modelling, where quantum statistical memory -- the memory required to model a stochastic process using a class of quantum machines -- can be used to quantify the structure of a stochastic process. By viewing ECA rules as transformations of stochastic patterns, we ask: Does an ECA generate structure as quantified by the quantum statistical memory, and if so, how quickly? We illustrate how the growth of this measure over time correctly distinguishes simple ECA from complex counterparts. Moreover, it provides a more refined means for quantitatively identifying complex ECAs -- providing a spectrum on which we can rank the complexity of ECA by the rate in which they generate structure.