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Completeness results for graphical quantum process languages

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From Feynman diagrams via Penrose graphical notation to quantum circuits, graphical languages are widely used in quantum theory and other areas of theoretical physics. The category-theoretical approach to quantum mechanics yields a new set of graphical languages, which allow rigorous and intuitive pictorial reasoning about quantum systems and processes. One such language is the ZX-calculus, which is built up of elements corresponding to maps in the computational and the Hadamard basis. We show that this graphical language is complete for stabilizer quantum mechanics and for the single-qubit Clifford+T group. This means that within those subtheories, any equality that can be derived using matrices can also be derived graphically. The ZX-calculus can thus be applied to a wide range of problems in quantum information and quantum foundations, from the analysis of quantum non-locality to the verification of measurement-based quantum computation and error-correcting codes. We also show how to construct a ZX-like graphical calculus for Spekkens’ toy bit theory, a local hidden variable theory which is nevertheless very similar to stabilizer quantum mechanics, and give its associated completeness proof. Hence Spekkens’ toy bit theory and stabilizer quantum mechanics—which is non-local—can be analysed and compared entirely graphically.

This talk is part of the CQIF Seminar series.

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