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Parafermions are Zn generalisations of Majorana quasiparticles, withfractional non-Abelian statistics. They can be used to encode topologicalqudits and perform Clifford operations by their braiding. We study the simplestcase of the Z3 parafermion chain and investigate the form of thenon-topological gate that arises through direct short-range interaction of theparafermion edge modes. We show that such an interaction gives rise to adynamical phase gate on the encoded ground space, with the strongest order ofthe interaction generating a non-Clifford gate which can be tuned to belong toeven levels of the Clifford hierarchy. We also illustrate the accessibility ofhighly non-contextual states using this dynamical gate. Finally, we propose anexperiment that simulates the braiding and dynamical evolutions of the Z3topological states with Rydberg atom technology.

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