Superintegrable cellular automata and dual unitary gates from Yang-Baxter maps

TL;DR

This paper introduces a class of one-dimensional cellular automata based on Yang-Baxter maps, revealing their superintegrability, ballistic charge propagation, and rich physical behaviors, including dual unitary gate structures and particle scattering.

Contribution

It demonstrates that Yang-Baxter map-based cellular automata are superintegrable with extensive conserved charges and explores their connection to dual unitary gates and complex transport phenomena.

Findings

Systems exhibit exponential conserved charges with ballistic propagation.

Models show coexistence of ballistic and diffusive transport.

Connection established between Yang-Baxter maps and dual unitary gates.

Abstract

We consider one dimensional block cellular automata, where the local update rules are given by Yang-Baxter maps, which are set theoretical solutions of the Yang-Baxter equations. We show that such systems are superintegrable: they possess an exponentially large set of conserved local charges, such that the charge densities propagate ballistically on the chain. For these quantities we observe a complete absence of "operator spreading". In addition, the models can also have other local charges which are conserved only additively. We discuss concrete models up to local dimensions $N\le 4$, and show that they give rise to rich physical behaviour, including non-trivial scattering of particles and the coexistence of ballistic and diffusive transport. We find that the local update rules are classical versions of the "dual unitary gates" if the Yang-Baxter maps are non-degenerate. We discuss consequences of dual unitarity, and we also discuss a family of dual unitary gates obtained by a non-integrable quantum mechanical deformation of the Yang-Baxter maps.