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Questions de localisabilité pour le calcul distribué

Abstract : This thesis is divided in two parts. Its starting point is the concept of resistance to localisation, an important concept in distributed quantum computing.In the first, theoretical part of this thesis, we go over the history of certain concepts and results in quantum information theory and distributed computing, such as the phenomenon of entanglement and the non-signalling condition in the first domain, and the LOCAL model and the colouring problem in the second domain. We then focus on the φ-LOCAL model, whose goal is to study the possibility of quantum distributed algorithms, and which was developedin 2009 by adapting the non-signalling condition to the LOCAL model. We introduce the concepts of global and local consistency in order to emphasise some shortcomings of this model. Finally, we present a more adequate version ofthe φ-LOCAL model.The second part of this thesis contains our major technical results in probability theory. We define the concept of k-localisability which is a probabilistic translation of the φ-LOCAL model. We show that this concept is close to but weaker than the concept of k-dependence which is well-studied in the probabilistic literature. We mention recent results concerning 1-dependent colouring of the path graph and the conclusion they allow us to reach with regards to 1-localisable colouring of the path graph : that it is possible with four or more colours. The rest of this part is dedicated to answering the question of the possibility of 1-localisable colouring of the path graph using three colours which we will show to be impossible. In answering this question we have made use of methods in linear programming and combinatorics. In particular, we prove a theorem on the explicit solution of a linear programming problem having a certain form, and a formula for the Catalan numbers.
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Submitted on : Friday, January 31, 2020 - 2:03:15 PM
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Ghazal Kachigar. Questions de localisabilité pour le calcul distribué. Combinatoire [math.CO]. Université de Bordeaux, 2019. Français. ⟨NNT : 2019BORD0339⟩. ⟨tel-02462588⟩



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