Departmental Colloquium
Lou Kauffman
UIC
Spin Networks, Quantum Topology and Topological Quantum Computing
Abstract: In the 1960's Roger Penrose discovered a remarkably lucid and
fundamentally topological diagrammatic method for handling the mathematics
of quantum particles with spin a multiple of 1/2. Calling this method
"spin networks" he proposed to reconstruct space-time from
a world of spin-exchange processes "pior" to space and time.
Penrose proved a Spin-Geometry Theorem that
reconstructed directions in a three-dimensional space from the abstract
networks. SpaceTime remained a problem for the theory.
In the 1980's the speaker discovered a topological generalization
of the Penrose spin networks that included a new knot invariant, the
Jones polynomnial. One can construct these q-deformed spin nets
on purely topological grounds, and they form models for topological
quantum field theories and specific models for the colored Jones
polynomials and the Witten-Reshetikhin Invariants of three manifolds.
The q-deformed spin networks form a bridge between combinatorial and
quantum field theoretic approaches to these invariants.
In the last few years Michael Freedman, Alexei Kitaev and their co-workers
showed that quantum computing can, in principle, be universally performed
within a
topological quantum field theory. This means that certain topological
quantum field theories have a rich enough structure to support unitary
representations of the braid group that are dense in the unitary groups.
A quantum computer is a unitary transformation that can be applied to a
prepared quantum state in such a way that measuring the result of the
transformation yields computational or algorithmic information.
In the 1990's Peter Shor showd that quantum computer's can factorize
integers faster than classical algorithms.
This talk will explain how the q-deformed spin nets and the bracket
model of the Jones polynomial provide a simple and accessible
construction of unitary braid representations that are universal for
quantum computation. We will begin with the Penrose spin nets, then
generalize them to include the Jones polynomial, then construct
the promised representations. We will then discuss how these representations
are related to the dream of physically realized quantum computing via the
possibilities inherent in the physics of two-dimensional media.
Friday March 9, 2007 at 3:00 PM in SEO 636