Hamza Jaffali : "Entanglement in Quantum Algorithms : Geometrical Approach and Related Tools"

PhD Works of Hamza JAFFALI : « Entanglement in Quantum Algorithms : Geometrical Approach and Related Tools ».

Abstract : Quantum entanglement is one of the most interesting phenomenon in Quantum Mechanics, and especially in Quantum Information. It is a fundamental resource in Quantum Computing, and its role in the efficiency and accuracy of quantum algorithms or protocols is not yet fully understood. In this thesis, we study quantum entanglement of multipartite states, and more precisely the nature of entanglement involved in quantum algorithms. This study is theoretical, and uses tools mainly coming from algebraic geometry. We focus on Grover’s and Shor’s algorithms, and determine what entanglement classes are reached (or not) by these algorithms, and this is the qualitative part of our study. Moreover, we quantitatively measure entanglement, using geometric and algebraic measures, and study its evolution through the several steps of these algorithms. We also propose original geometrical interpretations of the numerical results. On another hand, we also develop and exploit new tools for measuring, characterizing and classifying quantum entanglement. First, from a mathematical point of view, we study singularities of hypersurfaces associated to quantum states in order to characterize several entanglement classes. Secondly, we propose new candidates for maximally entangled states, especially for symmetric and fermionic systems, using polynomial invariants and geometric measure of entanglement. Finally, we use Machine Learning, more precisely the supervised approach using neural networks, to learn how to recognize algebraic varieties directly related with some entanglement classes.

Jury composition :

HOLWECK Frédéric (UTBM), et MEROLLA Jean-Marc (Femto-ST), Directeurs de thèse

PERDRIX Simon - Chargé de Recherche - Loria (reporter)

SAFEY EL DIN Mohab - Professeur - Université de Sorbonne (reporter)

COUTURIER Raphaël - Professeur - Université de Franche-Comté (reviewer)

FRANZ Uwe - Professeur - Université de Franche-Comté (reviewer)

JAUSLIN Hans-Rudolf - Professeur - Université de Bourgogne (reviewer)

LUQUE Jean-Gabriel - Professeur - Université de Rouen (reviewer)

Localization : Amphithéâtre I102, Bâtiment I (Accueil), Université de Technologie de Belfort-Montbéliard, Site de Belfort, 12 Rue Thierry Mieg - Belfort