Quantum-inspired identification of complex cellular automata

TL;DR

This work asks: Does an ECA generate structure as quantified by the quantum statistical memory, and can this be used to identify complex cellular automata, and illustrates how the growth of this measure over time correctly distinguishes simple ECA from complex counterparts.

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.

Figures (9)

• Figure 1: (a) The state of each cell in an ECA takes on a binary value, and is deterministically updated at each timestep according to the current states of it and its nearest neighbours. A rule number codifies this update by converting a binary representation of the updated states to decimal. Shown here is Rule 26 $(00011010)_2$. (b) The evolution of an ECA can be visualised via a two-dimensional grid of cells, where each row corresponds to the full state of the ECA at a particular time, and columns the evolution over time. Depicted here are examples of ECA for each of Wolfram's four classes.
• Figure 2: A quantum model consists of a unitary operator $U$ acting on a memory state $|{\sigma_j}\rangle$ and blank ancilla $|{0}\rangle$. Measurement of the ancilla produces the output symbol, with the statistics of the modelled process realised through the measurement statistics.
• Figure 3: An ECA is evolved from random initial conditions. Treating the ECA state at each timestep as a stochastic process, we then infer the quantum statistical memory $C^{(t)}_q$ and classical statistical complexity $C^{(t)}_\mu$. By observing how these measures change over time, we are able to deduce the complexity of the ECA rule.
• Figure 4: Generation of finite-width ECA evolution with open boundaries via extended ECA with periodic boundaries.
• Figure 5: $C_q^{(t)}$ plots for a selection of rules with longer $L$ and larger $t_{\text{max}}$. Plots shown for $(W=64,000, L=6)$, $(W=128,000, L=7)$, and $(W=256,000, L=8)$.