Abstract

Quantum picturalism' is a method for graphically representing processes in quantum mechanics. It is of particular interest for quantum information processes, defining a 'flow' of information through a protocol or algorithm. In this work I apply the category-theoretic work of Abramsky and Coecke to the topological cluster-state model of quantum computing to show the topological equivalence of defect strands in the cluster state and the graphical flow of the red/green calculus. I will concentrate on the pictorial representation, and use a minimal amount of the machinery of category theory. I give the equivalence between the graphical and topological information flows, and show the applicable rewrite algebra for this computing model. Finally I will show how the use of quantum picturalism gives a proof algebra for topological cluster state computing, from which we can derive previously unknown properties of the model. This work not only demonstrates for the first time a concrete realisation of quantum diagrammatics, but also gives us a native graphical language for the design and analysis of topological quantum algorithms.