Universality of Z3 parafermions via edge-mode interaction and quantum simulation of topological space evolution with Rydberg atoms

Parafermions are Zn generalizations of Majorana quasiparticles, with fractional non-Abelian statistics. They can be used to encode topological qudits and perform Clifford operations by their braiding. Here we investigate the generation of quantum gates by allowing Z3 parafermions to interact in order to achieve universality. In particular, we study the form of the nontopological gate that arises through direct short-range interaction of the parafermion edge modes in a Z3 parafermion chain. We show that such an interaction gives rise to a dynamical phase gate on the encoded ground space, generating a non-Clifford gate which can be tuned to belong to even levels of the Clifford hierarchy. We illustrate how to access highly noncontextual states using this dynamical gate. Finally, we propose an experiment that simulates the braiding and dynamical evolutions of the Z3 topological states with Rydberg atom technology.