Recent advances in quantum computational science promise substantial improvements in the speed with which certain classes of problems can be computed. Various algorithms that utilize the distinctively non-classical characteristics of quantum mechanics have been formulated to take advantage of this promising new approach to computation. One such algorithm was formulated by David Deutsch and Richard Jozsa. By measuring the output of a quantum network that implements this algorithm, it is possible to determine with N 1 measurements certain global properties of a function f(x), where N is the number of network inputs. Classically, it may not be possible to determine these same properties without evaluating f(x) a number of times that rises exponentially as N increases. Hitherto, the potential power of this algorithm has been explored in the context of qubits, the quantum computational analogue of classical bits. However, just as one can conceive of classical computation in the context of non-binary logic, such as ternary or quaternary logic, so also can one conceive of corresponding higher-order quantum computational equivalents.<p>This thesis investigates the behaviour of the Deutsch-Jozsa algorithm in the context of these higher-order quantum computational forms of logic and explores potential applications for this algorithm. An important conclusion reached is that, not only can the Deutsch-Jozsa algorithms known computational advantages be formulated in more general terms, but also a new algorithmic property is revealed with potential practical applications.

Recent advances in quantum computational science promise substantial improvements in the speed with which certain classes of problems can be computed. Various algorithms that utilize the distinctively non-classical characteristics of quantum mechanics have been formulated to take advantage of this promising new approach to computation. One such algorithm was formulated by David Deutsch and Richard Jozsa. By measuring the output of a quantum network that implements this algorithm, it is possible to determine with N 1 measurements certain global properties of a function f(x), where N is the number of network inputs. Classically, it may not be possible to determine these same properties without evaluating f(x) a number of times that rises exponentially as N increases. Hitherto, the potential power of this algorithm has been explored in the context of qubits, the quantum computational analogue of classical bits. However, just as one can conceive of classical computation in the context of non-binary logic, such as ternary or quaternary logic, so also can one conceive of corresponding higher-order quantum computational equivalents.<p>This thesis investigates the behaviour of the Deutsch-Jozsa algorithm in the context of these higher-order quantum computational forms of logic and explores potential applications for this algorithm. An important conclusion reached is that, not only can the Deutsch-Jozsa algorithms known computational advantages be formulated in more general terms, but also a new algorithmic property is revealed with potential practical applications.

Cette thèse a pour première vocation d'être un état de l'art sur le calcul quantique, sinon exhaustif, simple d'accès (chapitres 1, 2 et 3). La partie originale de cet essai consiste en deux approches mathématiques du calcul quantique concernant quelques systèmes quantiques : la première est de nature algébrique et fait intervenir des structures particulières : les corps et les anneaux de Galois (chapitre 4), la deuxième fait appel à la géométrie dite projective (chapitre 5). Cette étude a été motivée par le théorème de Kochen et Specker et par les travaux de Peres et Mermin qui en ont découlé

Cette thèse a pour première vocation d’être un état de l’art sur le calcul quantique, sinon exhaustif, simple d’accès (chapitres 1, 2 et 3). La partie originale de cet essai consiste en deux approches mathématiques du calcul quantique concernant quelques systèmes quantiques : la première est de nature algébrique et fait intervenir des structures particulières : les corps et les anneaux de Galois (chapitre 4), la deuxième fait appel à la géométrie dite projective (chapitre 5). Cette étude a été motivée par le théorème de Kochen et Specker et par les travaux de Peres et Mermin qui en ont découlé / The first vocation of this thesis would be a state of the art on the field of quantum computation, if not exhaustive, simple access (chapters 1, 2 and 3). The original (interesting) part of this treatise consists of two mathematical approaches of quantum computation concerning some quantum systems : the first one is an algebraic nature and utilizes some particular structures : Galois fields and rings (chapter 4), the second one calls to a peculiar geometry, known as projective one (chapter 5). These two approaches were motivated by the theorem of Kochen and Specker and by work of Peres and Mermin which rose from it