Representing modular tensor categories: a computer algebra system for topological quantum computing

Thesis

Topological quantum computation (TQC) is a new fault-tolerant approach to quantum information, where computations are carried out by braiding particles called anyons. Anyons are quasiparticles that exist in 2 + 1 dimensions and are neither bosons or fermions. Modular tensor categories capture the structure of anyon systems and thus serve as models for topological quantum computation. However, program of using category theory to find higher-level structures and protocols has yet to be applied to TQC due to the difficulty of working with modular tensor categories. This difficulty could be greatly mitigated by the development of a computer algebra system to represent such categories. Thus, a computer algebra system for representing modular tensor categories within the symmetric monoidal 2-category 2Vect was developed.

A general representation for 2Vect is described. This involves extending basic linear algebra operations to handle matrices of zero-dimension and 2-matrices (matrices of matrices). Then, representations for the 0-cells, 1-cells, and 2-cells of 2Vect are found. Due to the chosen representation, various structural isomorphisms are required to ensure equations expected of 1-categories hold in the 2-categories setting. These structural isomorphisms are explicitly constructed.

In order to identify special categories, such as modular tensor categories, a diagrammatic language of monoidal categories is extended for 2-categories. This language is used to construct the axioms for these special categories within 2Vect. Finally, a way of implementing these axioms within 2Vect in the computer algebra system is described. This system will now be used for the investigation of higher-level structures within modular tensor categories to better understand the structure of topological quantum computing.

Authors: Roberts, D

• Publication date: 2011
• Type of award: MSc by Research
• Level of award: Masters
• Awarding institution: Oxford University, UK
• Language: English
• Keywords: physics; quantum computing; modular tensor categories; anyons; computer science; category theory; mathematics; topological quantum computing; 2-Categories
• Subjects: Computer science (mathematics); Quantum theory (mathematics); Theoretical physics; Physical Sciences; Condensed matter theory; Mathematics; Physics