A categorical semantics for topological quantum computation

Abstract

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.