Feynman Diagrams and the Semantics of Quantum Computation

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Feynman diagrams were developed as a tool to show how particles interact, but similar diagrams describe the the flow of information in other quantum systems. Recent work has shown how such diagrams can describe the information flow in quantum computation. The first aspect of this project will be to create a universal quantum programming language called the 'quantum lambda calculus'. We hope to show that all other quantum programming languages can be translated into this one. Moreover, any program written in this language can be translated into a diagram of the above sort. These diagrams will give a 'semantics' for quantum programming languages that is, a systematic method for studying what programs in these languages actually do. The second aspect of this research will be a study within this framework of the 'topological quantum computers' currently under development. Such computers are based on the fractional quantum Hall effect, which allows quasiparticles called 'anyons' to form on a thin superconducting film in a magnetic field. Diagrammatic methods are perfectly suited to describing such computers. The broader implications of this workare that the proposed work should also contribute to understanding the links with the ideas in topological quantum field theories, and other mathematical models arising from the work on quantum gravity; and moreover, should yield insights on the promising current work on topological quantum computing