  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.
