An Abstract Approach to Entanglement in Quantum Computing

The project is fundamental research into quantum computation. Quantum computation is a new model of information processing which uses quantum physics to achieve computational power impossible for a conventional computer. Many theoretical models have been proposed for how such a computer should work, but most have been very close to the machine itself, rather than exposing the phenomena a programmer should take advantage of in order to make best use of her quantum computer. We propose to investigate structures found in quantum computing which may play a role in facilitating high level programming languages in a quantum computer. Our central interest is the role played by (multiparty) entanglement. It has been shown that increasing ammounts of entanglement in a quantum system are responsible for the predicted efficiency improvements of quantum algorithms over their classical counter parts. It is also known that sharing an entangled state between several agents allows those agents to carry out communications protocols which are impossible by conventional means. Most importantly, the manipulation of entanglement is fundamental in the various mesurement based quantum computing frameworks, most strikingly in the One-Way Model, where the entire computation is carried out by performing local measurements on a very large enangled state. We aim to expose the underlying structure which is responsible for these phenomena using the categorical approach developed by Abramsky and Coecke, further expanded in the author's thesis and joint work with Abramsky. This involves viewing each entangled state as a generalised channel between participants which computes a given function. There is an intimate connection between an entangled state and the process used to prepare. From here the structure of large entangled states can be computed simply as generalised theory of composable functions. Work developed in the author's thesis shows that entangled states can be represented as directed graphs within a certain category. We intend to develop the connection between the topological features of these graphs and the information processing capabilities of the quantum states they represent. Such a theory will open the door, not just to a greater understanding of quantum states, but to new languages for reasoning about them -- in particular new programming languages for defining quantum algorithms and protocols. Since this work is carried out in a rigourous categorical framework all components will be mechanisable, to produce type inference algorithms and other tools to aid future quantum informaticians.