Spin Networks, Anyonic Topological Quantum Computing and Quantum Gravity

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 $1 + 1 \rightarrow 0$ or $1$. The corresponding recoupling theory is intricate. The braiding is non-trivial and can model quantum computation.) The purpose of this talk is to give a simple model for the Fibonacci Anyons in terms of $q = e^{i Pi/5}$ deformed spin networks, and to show how the structure of the model proceeds from the structure of the bracket model of the Jones polynomial and the Dubrovnik (Kauffman) skein polynomial.

The point of view of this talk allows discussion of the relatiohship of quantum information theory and quantum computing with the knot polynomials. The use of spin networks in these models suggests a deeper dialogue with quantum gravity. In the first place, the result about Fibonacci anyons shows that a deformation of the classical spin networks to a nearby root of unity allows the generation of arbitrarily good approximations to unitary group transformation in the braiding representations associated with these spin networks. This means that at the mathematical level there is a unification of the generation of background space (spin geometry) and the generation of quantum mechanical evolutions. In a sense this is a background for a possible unified quantum geometry. Secondly there is the possiblity of using the machinery of the Temperley Lieb recoupling theory at a categorified level, using the framework of Khovanov homology, to generate new four dimensional hybrid algebraic/state sum models that may impinge on topological approaches to quantum gravity. All of this part of the research is in a state of flux, but we will report on its present state at the time of the talk.