exchange statistics. We call these anyons Fock parafermions, because they are the particles naturally associated to the parafermionic zero-energy modes, potentially realizable in mesoscopic arrays of fractional topological insulators. Josephson junctions modeled with Fock parafermions may display a

2\ensuremath{\pi}p

periodic relation between the Josephson current and the phase difference across the junction. Self-dual representations can be realized in terms of local quadratic combinations of either parafermions or Fock parafermions, an important requisite for physical implementation of quantum logic gates. The second-quantization description of Fock parafermions entails the concept of Fock algebra, i.e., a Fock space endowed with a statistical multiplication that captures and logically correlates these anyons' exclusion and exchange statistics. As a consequence normal ordering continues to be a well-defined operation.

DOI: 10.1103/physreva.89.012328

Related papers

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arXiv:hep-th/0307168Journal of Mathematical PhysicsOct 1, 200425 citations

p=2,3,\ensuremath{\cdots}

exclusion and

\ensuremath{\theta}=2\ensuremath{\pi}/p

2\ensuremath{\pi}p

{\mathbb{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.

Parafermion modes are non-Abelian anyons which were introduced as

{\mathbb{Z}}_{N}

generalizations of

{\mathbb{Z}}_{2}

Majorana states. In particular,

{\mathbb{Z}}_{3}

parafermions can be used to produce Fibonacci anyons, laying a path towards universal topological quantum computation. Due to their fractional nature, much of the theoretical work on

{\mathbb{Z}}_{3}

parafermions has relied on bosonization methods or parafermionic quasiparticles. In this paper, we introduce a representation of

{\mathbb{Z}}_{3}

parafermions in terms of purely fermionic models. We establish the equivalency of a family of lattice fermionic models written in the basis of the $t\ensuremath{-}J$ model with a Kitaev-like chain supporting free

{\mathbb{Z}}_{3}

parafermionic modes at its ends. By using density matrix renormalization group calculations, we are able to characterize the topological phase transition and study the effect of local operators (doping and magnetic fields) on the spatial localization of the parafermionic modes and their stability. Moreover, we discuss the necessary ingredients towards realizing

{\mathbb{Z}}_{3}

parafermions in strongly interacting electronic systems.

\mathbb{Z}_3

parafermion fractional quantum Hall state in the second Landau level with filling factor

ν=12/5

. These anyons are non-Abelian and are known to be capable of universal topological quantum computation. The quantum information is encoded in the fusion channels of pairs of such non-Abelian anyons and is protected from noise and decoherence by the topological properties of these systems.The quantum gates are realized by braiding of these anyons. We propose here an implementation of the

n

-qubit topological quantum register in terms of

2n+2

Fibonacci anyons. The matrices emerging from the anyon exchanges, i.e. the generators of the braid group for one qubit are derived from the coordinate wave functions of a large number of electron holes and 4 Fibonacci anyons which can furthermore be represented as correlation functions in

\mathbb{Z}_3

parafermionic two-dimensional conformal field theory. The representations of the braid groups for more than 4 anyons are obtained by fusing pairs of anyons before braiding, thus reducing eventually the system to 4 anyons.