Topological quantum computing and SU (2) braid group representations

2008 ◽  
Author(s):  
Louis H. Kauffman ◽  
Samuel J. Lomonaco, Jr.
Keyword(s):  
Quantum Computing ◽  
Braid Group ◽  
Group Representations ◽  
Topological Quantum

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 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.

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.

scholarly journals Finiteness for mapping class group representations from twisted Dijkgraaf–Witten theory

2018 ◽  
Vol 27 (06) ◽  
pp. 1850043 ◽  
Author(s):  
Paul P. Gustafson
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Braid Groups ◽  
Fusion Category ◽  
Mapping Class ◽  
Group Representations ◽  
Colored Graphs ◽  
Finite Image ◽  
Spanning Set

We show that any twisted Dijkgraaf–Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof et al. showing that the braid group images are finite [P. Etingof, E. C. Rowell and S. Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008)(1) 33–42]. In particular, our result answers their question regarding finiteness of images of arbitrary mapping class group representations in the affirmative. Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface. To do this translation, we use the fact that any twisted Dijkgraaf–Witten representation associated to a finite group [Formula: see text] and 3-cocycle [Formula: see text] is isomorphic to a Turaev–Viro–Barrett–Westbury (TVBW) representation associated to the spherical fusion category [Formula: see text] of twisted [Formula: see text]-graded vector spaces. The representation space for this TVBW representation is canonically isomorphic to a vector space of [Formula: see text]-colored graphs embedded in the surface [A. Kirillov, String-net model of Turaev-Viro invariants, Preprint (2011), arXiv:1106.6033 ]. By analyzing the action of the Birman generators [J. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–242] on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image.

scholarly journals Topological Quantum Computing and 3-Manifolds

2021 ◽  
Vol 3 (1) ◽  
pp. 153-165
Author(s):  
Torsten Asselmeyer-Maluga
Keyword(s):  
Quantum Computing ◽  
Berry Phase ◽  
Basic Structure ◽  
Quantum States ◽  
Point Of View ◽  
Surface Topology ◽  
Usual Sense ◽  
Closed Curves ◽  
Topological Quantum ◽  
2D System

In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. Today, the basic structure is a 2D system to realize anyons with braiding operations. From the topological point of view, we have to deal with surface topology. However, usual materials are 3D objects. Possible topologies for these objects can be more complex than surfaces. From the topological point of view, Thurston’s geometrization theorem gives the main description of 3-dimensional manifolds. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere. The whole system depends strongly on the topology of this complement, which is determined by non-contractible, closed curves. Every curve gives a contribution to the quantum states by a phase (Berry phase). Therefore, the quantum states can be manipulated by using the knot group (fundamental group of the knot complement). The universality of these operations was already showed by M. Planat et al.

scholarly journals Who Needs Category Theory?

10.29007/4dr3 ◽  
2020 ◽  
Author(s):  
Andreas Blass ◽  
Yuri Gurevich
Keyword(s):  
Quantum Computing ◽  
Category Theory ◽  
Contentious Issue ◽  
Topological Quantum ◽  
Mathematical Applications

In mathematical applications, category theory remains a contentious issue, with enthusiastic fans and a skepticalmajority. In a muted form this split applies to the authors ofthis note. When we learned that the only mathematically soundfoundation of topological quantum computing in the literature isbased on category theory, the skeptical author suggested to "decategorize" the foundation. But we discovered, to our surprise, thatcategory theory (or something like it) is necessary for the purpose,for computational reasons. The goal of this note is to give a high-level explanation of that necessity, which avoids details and whichsuggests that the case of topological quantum computing is farfrom unique.

scholarly journals Hurwitz Spaces and Braid Group Representations

2004 ◽  
Vol 34 (3) ◽  
pp. 1005-1030
Author(s):  
Eric P. Klassen ◽  
Yaacov Kopeliovich
Keyword(s):  
Braid Group ◽  
Group Representations ◽  
Hurwitz Spaces
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