Quantum-inspired identification of complex cellular automata

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.

Enhancing quantum models of stochastic processes with error mitigation

“…Comparatively, it was shown that Cq → C q if L ≥ R and when the length of the stochastic process goes to infinity, N → ∞. The quantum inference protocol was recently used to demonstrate the relative complexity of elementary cellular automate [46]. The cardinality of A implies that the quantum inference protocol produces a maximum of A L quantum memory states for L-length consecutive pasts.…”