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Topological Quantum Computing, Graphs, and Contextuality

Abstract

Operational errors or decoherence effects are natural barriers to quantum computing implementations. In classical computation, errors can be corrected using redundancy. The translation to the quantum case is not immediate, due to the non-cloning theorem. Quantum error correcting codes were found, since the 1990s. Initially by suitable adaptations of classical error correcting codes. The surface codes, or topological quantum error correction codes, introduced by Kitaev, are a special class of stabiliser codes. In this class, a code is given by a tessellation of a given two-dimensional manifold. The best known surface code is the toric code. Results obtained by the visitor founded a new technique to create topological quantum codes, generalising Kitaev codes and Bombs and Martin-Delgado techniques. Those codes, however, do not implement universal quantum computation. Quantum contextuality has been shown as the natural requirement to "magic state distillation", upgrading those Clifford operations to universal computation. This project aims to build new families of surface and coloured codes as well as the study of the relation between quantum contextuality and graphs over orientable and non-orientable surfaces. (AU)

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