Quantum computing with parafermions

Abstract

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.

Fermionized parafermions and symmetry-enriched Majorana modes

“…To address property (ii), we will first summarize nonAbelian braiding in the parafermion realization, which is known to be richer than in conventional Majorana systems [20][21][22][23][81][82][83]. Imagine four Z 4 parafermion zero modes α 1,...,4 realized at defects in a parent fractional-quantum-Hall fluid; see Fig.…”

Topological Phases of Parafermions: A Model with Exactly Solvable Ground States

“…TO survives weak perturbations and hosts indistinguishably weak or strong modes [36]. The importance of parafermionic zero-modes for topological quantum computation [37] motivates further investigations of these fractionalized systems.In this letter we provide a non-trivial family of parafermionic models for which the properties of the ground states can be exactly characterized. These models are gapped, display TO, have spatial-inversion and time-reversal symmetries, and feature weak edge modes; they thus belong to the same symmetry class for which weak edge modes have been discussed so far with numerical and perturbative analytical methods [28,31,36], with the advantage of being easy to handle.…”

“…TO survives weak perturbations and hosts indistinguishably weak or strong modes [36]. The importance of parafermionic zero-modes for topological quantum computation [37] motivates further investigations of these fractionalized systems.…”

How SU(2)$_4$ Anyons are Z$_3$ Parafermions

“…At the same time, there has also been increasing interest in so-called Z k parafermionic systems of Fradkin-Kadanoff-Fendley type [17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Such parafermions were introduced by Fradkin and Kadanoff [17] as mathematical operators which allowed for elegant solutions of certain two dimensional statistical models with Z k symmetry.…”

“…More recently it was realised that these same mathematical operators may be used to describe defects that can arise in certain topological models [18,[21][22][23][24][25][26][27][28][29][30][31] -and there have been several proposals to experimentally engineer such defects. In this context the defects can be thought of as particles having some variety of non-abelian braiding statistics.…”