Description

The aim of this project is to explore the topic of topological quantum computation. This will involve studying some general aspects of quantum computing for which Quantum Mechanics III or equivalent is a prerequisite. However, to understand some models for topological quantum computing requires quantum field theory, so Advanced Quantum Theory IV or equivalent is a recommended corequisite. Note that this project does not require previous knowledge of topology.

Quantum computers process quantum information and can, in principle, perform certain types of calculation much more efficiently than classical computers. A well-known example is the problem of factorising a product of two large prime numbers. The classical difficulty of this problem is the basis for the widely-used RSA encryption scheme, used e.g. for secure transmission over the internet. However, there are many very difficult challenges to constructing a practical quantum computer.

A significant problem when processing quantum information is that it is not possible to duplicate an arbitrary quantum state (the no-cloning theorem, i.e. there are no quantum photocopiers). This means that we cannot simply make multiple copies of quantum information before transmitting or processing it, and so cannot deal with errors (which will inevitably arise in any real system) by simply duplicating the data. However, there are more sophisticated ways of dealing with errors, and using topology is one method. The basic concept (which you will explore in detail) can be illustrated by considering some pieces of string, each with one end fixed to the floor, then wound around each other in some way, and the other ends then fixed to the ceiling. The idea is to encode and process the information in the way the strings are wound around each other. Noise which would usually introduce errors corresponds to moving the strings, but importantly (without actually breaking the strings) this cannot change the way they are wound, so this topological information is immune to such noise. The main idea of this project is to explore the way in which this can be realised using models which contain anyons which are a type of particle in two dimensions. The important issues are how the information is encoded, how it can be processed (i.e. how do we implement a quantum algorithm), and how robust is the system to potential errors. In practical terms there is also the crucial question of whether a theoretically viable model can actually be implemented.

You could explore issues such as comparing the potential computational power of classical vs quantum computers, and specifically whether or not topological quantum computers are theoretically equivalent to other types of quantum computers. Also, topological quantum computers are naturally suited to tasks such as calculating the Jones polynomial which arises in the problem of classifying knots, and this is something which could be explored in detail.

Prerequisites

Quantum Mechanics III or equivalent Physics module(s).

Corequisites

Advanced Quantum Theory IV or equivalent Physics module(s) introducing quantum field theory is recommended.

Resources

Good places to start are Quantiki and of course Wikipedia-Quantum Information and Wikipedia-Topological Quantum Computer. There are also excellent notes by John Preskill and useful links at his Quantum Computation webpage. There is also an accessible book Introduction to Topological Quantum Computation by Jiannis Pachos.