Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0909.1672 · Journal of Knot Theory and Its Ramifications

ANYONIC TOPOLOGICAL QUANTUM COMPUTATION AND THE VIRTUAL BRAID GROUP

Heather A. Dye, Louis Hirsch Kauffman

Published Jan 1, 2011• 3 citations
arXivHTMLPDF

Abstract

We introduce a recoupling theory for virtual braided trees. This recoupling theory can be utilized to incorporate swap gates into anyonic models of quantum computation.
DOI
10.1142/s0218216511008619

Related papers

Quantum Computing and Quantum Topology

Louis Hirsch Kauffman, Samuel J. Lomonaco

Abstract This paper is an introduction to relationships between quantum topology and quantum computing. We discuss unitary solutions to the YangBaxter equation that are universal quantum gates, quantum entanglement and topological entanglement, and we give an exposition of knot-theoretic recoupling theory, its relationship with topological quantum field theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten-Reshetikhin-Turaev invariant of three manifolds.

Chapman & Hall/CRC eBooks•Sep 19, 2007•0 citations

Topological quantum computation on supersymmetric spin chains

Indrajit Jana, Filippo Montorsi, Pramod Padmanabhan +1 more

Abstract Quantum gates built out of braid group elements form the building blocks of topological quantum computation. They have been extensively studied in SU(2)k quantum group theories, a rich source of examples of non-Abelian anyons such as the Ising (k = 2), Fibonacci (k = 3) and Jones-Kauffman (k = 4) anyons. We show that the fusion spaces of these anyonic systems can be precisely mapped to the product state zero modes of certain Nicolai-like supersymmetric spin chains. As a result, we can realize the braid group in terms of the product state zero modes of these supersymmetric systems. These operators kill all the other states in the Hilbert space, thus preventing the occurrence of errors while processing information, making them suitable for quantum computing.

arXiv:2209.03822•Journal of High Energy Physics•Feb 27, 2023•2 citations

GEOMETRIC QUANTUM COMPUTATION AND TOPOLOGICAL QUANTUM COMPUTATION——RECENT APPLICATION OF GLOBAL PROPERTIES OF QUANTUM SYSTEMS

Huazhong Li

The new field of recent developments in quantum computation is reviewed This is an elementary introduction to quantum geometric computation and topological quantum computation.

Progress in physics•Jan 1, 2008•0 citations

Spin Networks and Anyonic Topological Computing II

Louis Hirsch Kauffman, Samuel J. Lomonaco

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. The simplest case of these models is the Fibonacci model, itself universal for quantum computation. We here formulate these braid group representations in a form suitable for computation and algebraic work.

•Nov 21, 2012•0 citations

Spin networks and anyonic topological computing

Louis Hirsch Kauffman, Samuel J. Lomonaco

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups.

arXiv:quant-ph/0603131•Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•May 5, 2006•13 citations

Spin networks and anyonic topological computing II

Louis H. Kauffman, Samuel J. Lomonaco, Jr.

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. The simplest case of these models is the Fibonacci model, itself universal for quantum computation. We here formulate these braid group representations in a form suitable for computation and algebraic work.

arXiv:0707.3678•Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Apr 27, 2007•0 citations
Abstract This paper is an introduction to relationships between quantum topology and quantum computing. We discuss unitary solutions to the YangBaxter equation that are universal quantum gates, quantum entanglement and topological entanglement, and we give an exposition of knot-theoretic recoupling theory, its relationship with topological quantum field theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten-Reshetikhin-Turaev invariant of three manifolds.
Abstract Quantum gates built out of braid group elements form the building blocks of topological quantum computation. They have been extensively studied in SU(2)k quantum group theories, a rich source of examples of non-Abelian anyons such as the Ising (k = 2), Fibonacci (k = 3) and Jones-Kauffman (k = 4) anyons. We show that the fusion spaces of these anyonic systems can be precisely mapped to the product state zero modes of certain Nicolai-like supersymmetric spin chains. As a result, we can realize the braid group in terms of the product state zero modes of these supersymmetric systems. These operators kill all the other states in the Hilbert space, thus preventing the occurrence of errors while processing information, making them suitable for quantum computing.