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Spin Networks and Anyonic Topological Quantum Computing
by
Louis H. Kauffman
University of Illinois at Chicago
Coauthors: Samuel J. Lomonaco, UMBC
Spin networks were invented/discovered by Roger Penrose in an attempt to provide a combinatorial precursor to spacetime. In his Spin-Geometry Theorem, Penrose showed how angular properties of three dimensional space would emerge from self-interactions of large spin networks. The Penrose theory of spin networks eventually was generalized to a recoupling theory that began with the bracket polynomial skein relation rather than the Penrose binor identity. This q-deformed spin network theory has been of use in constructing SU(2)q topological quantum field theories, the Witten invariants of three manifolds, and measurement and spin-foam techniques in loop quantum gravity. Recently, Freedman, Kitaev and their collaborators have shown how braiding operators in certain topological quantum field theories are universal for quantum computation. In particular, one can focus on the topoloogical quantum field theory called Fibonacci Anyons: There are two basic particles call them 1 and 0. The only non-trivial interaction is 1 + 1 -- > 0, or 1. The corresponding recoupling theory is intricate. The braiding is non-trivial and can model quantum computation. One obtains unitary representations of the braid group that are dense in the corresponding unitary groups. The purpose of this talk is to give a simple model for the Fibonacci Anyons in terms of q = ei π/5 deformed spin networks, and to show how the structure of the model proceeds from the structure of the bracket model of the Jones polynomial. We will also discuss how these models are related to a specialization of the Durbrovnik polynomial, and how this specialization gives an invariant of virtual knots and links. The point of view of this talk allows discussion of the relationhship of quantum information theory and quantum computing with the Jones polynomial. The use of spin networks in these models suggests a deeper dialogue with quantum gravity.
Date received: February 7, 2005
Copyright © 2005 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # capo-26.