${\mathbb{Z}}_{d}$ 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.${\mathbb{Z}}_{d}$ 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.

arXiv:1511.02704Physical review. B./Physical review. BMar 2, 201670 citations

Generalized quantum statistics and Lie (super)algebras

Nedialka I. Stoilova

Generalized quantum statistics, such as paraboson and parafermion statistics, are characterized by triple relations which are related to Lie (super)algebras of type B. The correspondence of the Fock spaces of parabosons, parafermions as well as the Fock space of a system of parafermions and parabosons to irreducible representations of (super)algebras of type B will be pointed out. Example of generalized quantum statistics connected to the basic classical Lie superalgebra B(1|1) ≡ osp(3|2) with interesting physical properties, such as noncommutative coordinates, will be given. Therefore the article focuses on the question, addressed already in 1950 by Wigner: do the equation of motion determine the quantum mechanical commutation relation?Generalized quantum statistics, such as paraboson and parafermion statistics, are characterized by triple relations which are related to Lie (super)algebras of type B. The correspondence of the Fock spaces of parabosons, parafermions as well as the Fock space of a system of parafermions and parabosons to irreducible representations of (super)algebras of type B will be pointed out. Example of generalized quantum statistics connected to the basic classical Lie superalgebra B(1|1) ≡ osp(3|2) with interesting physical properties, such as noncommutative coordinates, will be given. Therefore the article focuses on the question, addressed already in 1950 by Wigner: do the equation of motion determine the quantum mechanical commutation relation?

arXiv:1512.05076AIP conference proceedingsJan 1, 20160 citations

Free field representation of parafermions and related coset models

Anton A. Gerasimov, Andrei Vladimirovich Marshakov, A. A. Morozov

Nuclear Physics BDec 1, 198972 citations

SUPERSYMMETRY OF PARAFERMIONS

SERGEI KLISHEVICH, Mikhail S. Plyushchay

We show that the single-mode parafermionic type systems possess supersymmetry, which is based on the symmetry of characteristic functions of the parafermions related to the generalized deformed oscillator of Daskaloyannis et al. The supersymmetry is realized in both unbroken and spontaneously broken phases. As in the case of parabosonic supersymmetry observed recently by one of the authors, the form of the associated superalgebra depends on the order of the parafermion and can be linear or nonlinear in the Hamiltonian. The list of supersymmetric parafermionic systems includes usual parafermions, finite-dimensional q-deformed oscillator, q-deformed parafermionic oscillator and parafermionic oscillator with internal Z2 structure.We show that the single-mode parafermionic type systems possess supersymmetry, which is based on the symmetry of characteristic functions of the parafermions related to the generalized deformed oscillator of Daskaloyannis et al. The supersymmetry is realized in both unbroken and spontaneously broken phases. As in the case of parabosonic supersymmetry observed recently by one of the authors, the form of the associated superalgebra depends on the order of the parafermion and can be linear or nonlinear in the Hamiltonian. The list of supersymmetric parafermionic systems includes usual parafermions, finite-dimensional q-deformed oscillator, q-deformed parafermionic oscillator and parafermionic oscillator with internal Z2 structure.

arXiv:hep-th/9905149Modern Physics Letters ADec 21, 199962 citations

parafermioniques, et leurs applications.

Benoit Estienne

Jan 1, 20090 citations

{\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.