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ZX-Calculi for Quantum Computing and their Completeness

Renaud Vilmart 1
1 MOCQUA - Designing the Future of Computational Models
Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
Abstract : The ZX-Calculus is a powerful and intuitive graphical language, based on category theory, that allows for quantum reasoning and computing. Quantum evolutions are seen in this formalism as open graphs, or diagrams, that can be transformed locally according to a set of axioms that preserve the result of the computation. One of the most important aspects of language is its completeness: Given two diagrams that represent the same quantum evolution, can I transform one into the other using only the graphical rules allowed by the language? If this is the case, it means that the graphical language captures quantum mechanics entirely. The language is known to be complete for a particular subclass (or fragment) of quantum evolutions, called Clifford. Unfortunately, this one is not universal: we cannot represent, or even approach, certain quantum evolutions. In this thesis, we propose to extend the set of axioms to obtain completeness for larger fragments of the language, which in particular are approximately universal, or even universal. To do this, we first use the completeness of another graphical language and transport this result to the ZX-Calculus. In order to simplify this tedious step, we introduce an intermediate language, interesting in itself as it captures a particular but universal fragment of quantum mechanics: Toffoli-Hadamard. We then define the notion of a linear diagram, which provides a uniform proof for some sets of equations. We also define the notion of singular value decomposition of a diagram, which allows us to avoid a large number of calculations. In a second step, we define a normal form that exists for an infinite number of fragments of the language, as well as for the language itself, without restriction. Thanks to this, we reprove the previous completeness results, but this time without using any third party language, and we derive new ones for other fragments. The controlled states, used for the definition of the normal form, are also useful for performing non-trivial operations such as sum, term-to-term product, or concatenation.
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Renaud Vilmart. ZX-Calculi for Quantum Computing and their Completeness. Logic in Computer Science [cs.LO]. Université de Lorraine, 2019. English. ⟨NNT : 2019LORR0130⟩. ⟨tel-02395443⟩

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