The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and 2D-magnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like $e^{-\ał}$, where
ł is a length scale, and
α is some positive constant. In contrast, the $\q$presumptive" qubit-model of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about
10−4) before computation can be stabilized.
The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and 2D-magnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like e − α ℓ e^{-\alpha \ell } , where ℓ \ell is a length scale, and α \alpha is some positive constant. In contrast, the “presumptive" qubit-model of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about 10 − 4 10^{-4} ) before computation can be stabilized.
The aim of this thesis is to develop an abstract categorical setup in order to show that C -colored manifolds (i.e. compact closed manifolds with boundary where each boundary component is colored with an object of a semisimple strongly ribbon category) behaves basically in a similar manner as quantum circuits under the action of a unitary modular functor. There, the set of gates is composed only of braid operations, rotations and Dehn-twists. We first introduce the basic mathematical structure of a quantum circuit. We then provide a complete development of a 2-dimensional CW-complex over an extended surface. Furthermore, 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. We then conclude by giving a complete 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.