Graphical languages offer intuitive and rigorous formalisms for quantum physics. They can be used to simplify expressions, derive equalities, and do computations. Yet in order to replace conventional formalisms, rigour alone is not sufficient: the new formalisms also need to have equivalent deductive power. This requirement is captured by the property of completeness, which means that any equality that can be derived using some standard formalism can also be derived graphically. In this thesis, I consider the ZX-calculus, a graphical language for pure state qubit quantum mechanics. I show that it is complete for pure state stabilizer quantum mechanics, so any problem within this fragment of quantum theory can be fully analysed using graphical methods. This includes questions of central importance in areas such as error-correcting codes or measurement-based quantum computation. Furthermore, I show that the ZX-calculus is complete for the single-qubit Clifford+T group, which is approximately universal: any single-qubit unitary can be approximated to arbitrary accuracy using only Clifford gates and the T-gate. In experimental realisations of quantum computers, operations have to be approximated using some such finite gate set. Therefore this result implies that a wide range of realistic scenarios in quantum computation can be analysed graphically without loss of deductive power. Lastly, I extend the use of rigorous graphical languages outside quantum theory to Spekkens' toy theory, a local hidden variable model that nevertheless exhibits some features commonly associated with quantum mechanics. The toy theory for the simplest possible underlying system closely resembles stabilizer quantum mechanics, which is non-local; it thus offers insights into the similarities and differences between classical and quantum theories. I develop a graphical calculus similar to the ZX-calculus that fully describes Spekkens' toy theory, and show that it is complete. Hence, stabilizer quantum mechanics and Spekkens' toy theory can be fully analysed and compared using graphical formalisms. Intuitive graphical languages can replace conventional formalisms for the analysis of many questions in quantum computation and foundations without loss of mathematical rigour or deductive power.

Employing correlation consistent basis sets coupled with electronic structure methods has enabled accurate predictions of chemical properties for second- and third-row main group and transition metal molecular species. For third-row (Ga-Kr) molecules, the performance of the correlation consistent basis sets (cc-pVnZ, n=D, T, Q, 5) for computing energetic (e.g., atomization energies, ionization energies, electron and proton affinities) and structural properties using the ab initio coupled cluster method including single, double, and quasiperturbative triple excitations [CCSD(T)] and the B3LYP density functional method was examined. The impact of relativistic corrections on these molecular properties was determined utilizing the Douglas-Kroll (cc-pVnZ-DK) and pseudopotential (cc-pVnZ-PP) forms of the correlation consistent basis sets. This work was extended to the characterization of molecular properties of novel chemically bonded krypton species, including HKrCl, FKrCF3, FKrSiF3, FKrGeF3, FKrCCF, and FKrCCKrF, and provided the first evidence of krypton bonding to germanium and the first di-krypton system. For second-row (Al-Ar) species, the construction of the core-valence correlation consistent basis sets, cc-pCVnZ was reexamined, and a revised series, cc-pCV(n+d)Z, was developed as a complement to the augmented tight-d valence series, cc-pV(n+d)Z. Benchmark calculations were performed to show the utility of these new sets for second-row species. Finally, the correlation consistent basis sets were used to study the structural and spectroscopic properties of Au(CO)Cl, providing conclusive evidence that luminescence in the solid-state can be attributed to oligomeric species rather than to the monomer.

La mécanique quantique est sans aucun doute la théorie la mieux vérifiée qui n’a jamais existée. En se retournant vers le passé, nous constatons qu’un siècle de théorie quantique a non seulement changé la perception que nous avons de l’univers dans lequel nous vivons mais aussi est responsable de plusieurs concepts technologiques qui ont le potentiel de révolutionner notre monde. La présente dissertation a pour but de mettre en avance ces potentiels, tant dans le domaine théorique qu’expérimental. Plus précisément, dans un premier temps, nous étudierons des protocoles de communication quantique et démontrerons que ces protocoles offrent des avantages de sécurité qui n’ont pas d’égaux en communication classique. Dans un deuxième temps nous étudierons trois problèmes spécifiques en clonage quantique ou chaque solution apportée pourrait, à sa façon, être exploitée dans un problème de communication quantique. Nous débuterons par décrire de façon théorique le premier protocole de communication quantique qui a pour but la distribution d’une clé secrète entre deux parties éloignées. Ce chapitre nous permettra d’introduire plusieurs concepts et outils théoriques qui seront nécessaires dans les chapitres successifs. Le chapitre suivant servira aussi d’introduction, mais cette fois-ci penché plutôt vers le côté expériemental. Nous présenterons une élégante technique qui nous permettra d’implémenter des protocoles de communication quantique de façon simple. Nous décrirons ensuite des expériences originales de communication quantique basées sur cette technique. Plus précisément, nous introduirons le concept de filtration d’erreur et utiliserons cette technique afin d’implémenter une distribution de clé quantique bruyante qui ne pourrait pas être sécurisé sans cette technique. Nous démontrerons ensuite des expériences implémentant le tirage au sort quantique et d’identification quantique. Dans un deuxième temps nous étudierons des problèmes de clonage quantique basé sur le formalisme introduit dans le chapitre d’introduction. Puisqu’il ne sera pas toujours possible de prouver l’optimalité de nos solutions, nous introduirons une technique numérique qui nous permettra de mettre en valeur nos résultats.

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