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Analysis of quantum hypergraph states in the ZH-calculus

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Abstract

In this dissertation, we prove the known quantum hypergraph state manipulation rules, and introduce metarules for showing the correctness of measurement-based quantum computing (MBQC) resource states and measurement patterns in the ZH-calculus. In particular, we study the action of local Pauli operators on quantum hypergraph states, prove the generalisation of local complementation laws for hypergraph states and derive hypergraph transformation rules for local measurements, which have never been studied in the graphical calculus before. We then apply the derived metarules to study the two hypergraph-based MBQC states: the Union Jack state and the Takeuchi-Morimae-Hayashi state. Our examples show that the use of graphical calculus yields simple and intuitive proofs; in particular the metarules provide a formal and straightforward way of validating MBQC measurement patterns.

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