Simulating non-Abelian statistics of parafermions with superconducting processor

Parafermions, which can be viewed as a fractionalized version of Majorana modes, exhibit profound non-Abelian statistics and emerge in topologically ordered systems, while their realization in experiment has been challenging. Here we propose a novel experimental scheme for the quantum simulation of parafermions and their non-Abelian braiding statistics in superconducting (SC) circuits by realizing the $\mathbb{Z}_d$ plaquette model on a two-dimensional lattice. Two protocols using quantum circuits and non-destructive measurements are introduced to prepare the topologically ordered ground state, on which parafermion pairs are created by engineering dislocations. We then propose a generalized code deformation approach to realize the fusion and non-Abelian braiding of parafermions, and show the application of this approach to the $\mathbb{Z}_3$ parafermions. We also examine the real experimental parameter regime to confirm the feasibility of our scheme in SC devices. This work extends previous quantum simulations of topological defects in SC qubits to qudit systems and opens up a promising way for parafermion-based high-dimensional topological quantum computing with experimental feasibility.