Robust Non-Clifford gate for Parafermions

Anyon models of Parafermions can be extended through the addition of a quantized fractional flux and made universal. Braiding of Parafermions provides the Clifford gates while the braiding of a quantized fractional flux around a pair of Parafermions can provide the requisite Non-Clifford gate needed for universality of quantum gates. For example, braiding of half fluxon with h/4e quantized flux around Z_N(N>2) parafermion pair can provide the Non-Clifford Z^(1/2) gate, where Z is the logical qudit operator. Josephson junction made of spin-triplet superconductors can host a half fluxon which can be braided around parafermionic defects implementing the Z^(1/2) gate. We demonstrate the implementation of this Non-Clifford gate and investigate its robustness.