Anomalous periodicity and parafermion hybridization in superconducting qubits

Topological quantum computation relies on a protected degenerate subspace enabling complicated operations in a noise-resilient way. To this end, hybrid platforms based on non-Abelian quasiparticles such as parafermions hold great promise. These are predicted to emerge at the interface between fractional quantum Hall states and superconductors and therefore naturally couple to superconducting qubits. Here, we study a parafermionic fluxonium circuit and show that the presence of topological states yields a striking periodicity in the qubit spectrum. In addition, peculiar and marked signatures of different parafermion coupling, associated with multiple tunneling of fractional quasiparticles, can be detected in the qubit microwave spectrum. Finite parafermion coupling can reduce the full degeneracy of the non-Abelian manifold, and we show that this configuration can be used to assess the remaining degree of protection of the system.