Quantum computing with parafermions

Z(d) parafermions are exotic non-Abelian quasiparticles generalizing Majorana fermions, which correspond to the case d = 2. In contrast to Majorana fermions, braiding of parafermions with d > 2 allows one to perform an entangling gate. This has spurred interest in parafermions, and a variety of condensed matter systems have been proposed as potential hosts for them. In this work, we study the computational power of braiding parafermions more systematically. We make no assumptions on the underlying physical model but derive all our results from the algebraical relations that define parafermions. We find a family of 2d representations of the braid group that are compatible with these relations. The braiding operators derived this way reproduce those derived previously from physical grounds as special cases. We show that if a d-level qudit is encoded in the fusion space of four parafermions, braiding of these four parafermions allows one to generate the entire single-qudit Clifford group (up to phases), for any d. If d is odd, then we show that in fact the entire many-qudit Clifford group can be generated.