Braiding and Majorana fermions

2017 ◽  
Vol 26 (09) ◽  
pp. 1743001 ◽  
Author(s):  
Louis H. Kauffman
Keyword(s):  
Quantum Computing ◽  
Clifford Algebra ◽  
General Result ◽  
Braid Group ◽  
Clifford Algebras ◽  
Majorana Fermions ◽  
Group Representations ◽  
Topological Quantum

In this paper, we study unitary braid group representations associated with Majorana fermions. Majorana fermions are represented by Majorana operators, elements of a Clifford algebra. The paper proves a general result about braid group representations associated with Clifford algebras and compares this result with the Ivanov braiding associated with Majorana operators and with other braiding representations associated with Majorana fermions such as the Fibonacci model for universal topological quantum computing.

scholarly journals Majorana Fermions and representations of the braid group

2018 ◽  
Vol 33 (23) ◽  
pp. 1830023
Author(s):  
Louis H. Kauffman
Keyword(s):  
Clifford Algebra ◽  
General Result ◽  
Braid Group ◽  
Clifford Algebras ◽  
Majorana Fermions ◽  
Group Representations

In this paper we study unitary braid group representations associated with Majorana Fermions. Majorana Fermions are represented by Majorana operators, elements of a Clifford algebra. The paper recalls and proves a general result about braid group representations associated with Clifford algebras, and compares this result with the Ivanov braiding associated with Majorana operators. The paper generalizes observations of Kauffman and Lomonaco and of Mo-Lin Ge to show that certain strings of Majorana operators give rise to extraspecial 2-groups and to braiding representations of the Ivanov type.

scholarly journals Extraspecial two-Groups, generalized Yang-Baxter equations and braiding quantum gates

2010 ◽  
Vol 10 (7&8) ◽  
pp. 685-702
Author(s):  
E.C. Rowell ◽  
Y. Zhang ◽  
Y.-S. Wu ◽  
M.-L. Ge
Keyword(s):  
Quantum Computing ◽  
Braid Group ◽  
Quantum Error Correction ◽  
Unitary Representations ◽  
Quantum Gates ◽  
Group Representations ◽  
Unitary Evolution ◽  
Ghz States ◽  
Topological Quantum ◽  
Quantum Error

In this paper we describe connections among extraspecial 2-groups, unitary representations of the braid group and multi-qubit braiding quantum gates. We first construct new representations of extraspecial 2-groups. Extending the latter by the symmetric group, we construct new unitary braid representations, which are solutions to generalized Yang-Baxter equations and use them to realize new braiding quantum gates. These gates generate the GHZ (Greenberger-Horne-Zeilinger) states, for an arbitrary (particularly an \emph{odd}) number of qubits, from the product basis. We also discuss the Yang-Baxterization of the new braid group representations, which describes unitary evolution of the GHZ states. Our study suggests that through their connection with braiding gates, extraspecial 2-groups and the GHZ states may play an important role in quantum error correction and topological quantum computing.

Majorana fermions in condensed-matter physics

2016 ◽  
Vol 30 (19) ◽  
pp. 1630012 ◽  
Author(s):  
A. J. Leggett
Keyword(s):  
Quantum Computing ◽  
Current Literature ◽  
Particle Physics ◽  
Condensed Matter ◽  
Broken Symmetry ◽  
Majorana Fermions ◽  
Condensed Matter Physics ◽  
Late Professor ◽  
Topological Quantum ◽  
Abdus Salam

It is an honor and a pleasure to have been invited to give a talk in this conference celebrating the memory of the late Professor Abdus Salam. To my regret, I did not know Professor Salam personally, but I am very aware of his work and of his impact on my area of specialization, condensed matter physics, both intellectually through his ideas on spontaneously broken symmetry and more practically through his foundation of the ICTP. Since I assume that most of this audience are not specialized in condensed-matter physics, I thought I would talk about one topic which to some extent bridges this field and the particle-physics interests of Salam, namely Majorana fermions (M.F.s). However, as we shall see, the parallels which are often drawn in the current literature may be a bit too simplistic. I will devote most of this talk to a stripped-down exposition of the current orthodoxy concerning M.F.s. in condensed-matter physics and their possible applications to topological quantum computing (TQC), and then at the end briefly indicate why I believe this orthodoxy may be seriously misleading.

scholarly journals Z2 Topological Order and Topological Protection of Majorana Fermion Qubits

2021 ◽  
Vol 6 (1) ◽  
pp. 11
Author(s):  
Rukhsan Ul Haq ◽  
Louis H. Kauffman
Keyword(s):  
Braid Group ◽  
Transverse Field ◽  
Majorana Fermion ◽  
Chain Model ◽  
Topological Phases ◽  
Topological Order ◽  
Group Representations ◽  
Broad Perspective ◽  
Topological Quantum ◽  
Symmetry Operators

The Kitaev chain model exhibits topological order that manifests as topological degeneracy, Majorana edge modes and Z2 topological invariant of the bulk spectrum. This model can be obtained from a transverse field Ising model(TFIM) using the Jordan–Wigner transformation. TFIM has neither topological degeneracy nor any edge modes. Topological degeneracy associated with topological order is central to topological quantum computation. In this paper, we explore topological protection of the ground state manifold in the case of Majorana fermion models which exhibit Z2 topological order. We show that there are at least two different ways to understand this topological protection of Majorana fermion qubits: one way is based on fermionic mode operators and the other is based on anti-commuting symmetry operators. We also show how these two different ways are related to each other. We provide a very general approach to understanding the topological protection of Majorana fermion qubits in the case of lattice Hamiltonians. We then show how in topological phases in Majorana fermion models gives rise to new braid group representations. So, we give a unifying and broad perspective of topological phases in Majorana fermion models based on anti-commuting symmetry operators and braid group representations of Majorana fermions as anyons.

scholarly journals GENERALIZED YANG–BAXTER EQUATIONS AND BRAIDING QUANTUM GATES

2012 ◽  
Vol 21 (09) ◽  
pp. 1250087 ◽  
Author(s):  
REBECCA S. CHEN
Keyword(s):  
Braid Group ◽  
Knot Theory ◽  
Information Science ◽  
Quantum Gates ◽  
Quantum Computers ◽  
Baxter Equation ◽  
Group Representations ◽  
Ongoing Research ◽  
Topological Quantum ◽  
Mathematics And Physics

Solutions to the Yang–Baxter equation — an important equation in mathematics and physics — and their afforded braid group representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum information science. In particular, unitary representations of the braid group are desired because they generate braiding quantum gates. These are actively studied in the ongoing research into topological quantum computing. A generalized Yang–Baxter equation was proposed a few years ago by Eric Rowell et al. By finding solutions to the generalized Yang–Baxter equation, we obtain new unitary braid group representations. Our representations give rise to braiding quantum gates and thus have the potential to aid in the construction of useful quantum computers.

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