2 November 2004
4:00 - 5:30   Eric Paquette
Towards A Categorical Semantics For Topological Quantum Computing

Abstract:
We show that C-colored manifolds (i.e. compact closed manifolds with
boundary where each boundary component is colored with an object of a
semisimple ribbon category) behaves in a similar manner as quantum
circuits under the action of a unitary modular functor. There, the set
of elementary gates is composed only of braid operations, rotations and
Dehn-twists.

We introduce the basic mathematical structure of a quantum circuit. We
then provide a complete development of a 2-dimensional CW-complex over
the markings of extended surfaces, where the later is connected and
simply-connected. Next, we provide a complete development of the
categorical framework in order to construct a C-extended unitary modular
functor (UMF) acting from the category of C-colored surfaces and
morphisms of C-colored surfaces to the category of finite-dimensional
vector spaces and linear isomorphisms.

From there, we conclude by giving a complete categorical semantics for
topological quantum computation including an abstract version of the
inner product, basic data units, basic data transformations, projectors
and the notion of topological invariance of the algorithms.