Invariants in the II1 subfactor theory of von Neumann algebras

Thesis | January-November 2021

This was a year-long Honours project completed in 2021, in fulfilment of my Honours degree.

A II1 subfactor is an inclusion of two von Neumann algebras with trivial centre and equipped with a well-behaved trace. These objects have numerous connections with statistical mechanics, knot theory, quantum group theory, and conformal field theory.

My thesis is an account of the development of subfactor theory in the 1980s and 1990s, focusing on two major invariants which emerged in that time: the index and the principal graph. I give an exposition of two proofs of the Jones index theorem, a major result of Vaughan Jones: one proof using the theory of Temperley-Lieb algebras, and one proof leveraging the structure of the principal graph.

Optimal compilation of quantum algorithms

Report | February-June 2020

Quantum compilation is the task of approximating user-designed quantum algorithms with sequences of machine-executable quantum gates (operations). I used techniques from group theory and graph theory to model compilation strategies for quantum algorithms. I tested the efficiency of these strategies with the computer algebra system Sage, taking into account physical constraints such as decoherence and spatial separation of quantum bits.

A teaching model for single-qubit physics

Report | December 2018-February 2019

A qubit is a basic unit of a quantum computer. A nitrogen vacancy is a physical implementation of a qubit that is recognised for its viability at room temperature. I wrote a 30-page set of teaching notes on qubit physics and nitrogen vacancies for a potential future course at ANU. I also performed and documented a teaching experiment to demonstrate key properties of nitrogen vacancies.