<p>This thesis presents and develops the path integral simulation techniques in application to small quantum systems at finite temperatures. The first goal is to obtain exact thermodynamic expressions for systems of noninteracting</p><p>The rest and the major part of the thesis is dedicated to the development and testing of Bead-Fourier path integral molecular dynamics. Although, path integral molecular dynamics as well as path integral Monte Carlo are well</p><p>First, molecular dynamics under Bead-Fourier scheme was developed and tested on the examples of quantum harmonic oscillator and Hydrogen atom. The main attention was paid to ergodicity problems. Then we addressed the question,</p><p>Later, the formalism for identical particles was developed.</p><p>Finally, the question of molecular dynamics efficacy was raised. It was shown, that formalisms for identical and distinguishable particles, both, can be reformulated into a more efficient ones, providing all dynamical variables</p>

This thesis presents and develops the path integral simulation techniques in application to small quantum systems at finite temperatures. The first goal is to obtain exact thermodynamic expressions for systems of noninteracting The rest and the major part of the thesis is dedicated to the development and testing of Bead-Fourier path integral molecular dynamics. Although, path integral molecular dynamics as well as path integral Monte Carlo are well First, molecular dynamics under Bead-Fourier scheme was developed and tested on the examples of quantum harmonic oscillator and Hydrogen atom. The main attention was paid to ergodicity problems. Then we addressed the question, Later, the formalism for identical particles was developed. Finally, the question of molecular dynamics efficacy was raised. It was shown, that formalisms for identical and distinguishable particles, both, can be reformulated into a more efficient ones, providing all dynamical variables

Atoms in optical lattices are promising candidates to implement quantum information processing. Their behaviour is well understood on a microscopic level, they exhibit excellent coherence properties, and they can be easily manipulated using external fields. In very deep optical lattices, each atom is restricted to a single lattice site and can be used as a qubit. If the lattice is shallow enough such that the atoms can move, their properties can be used to simulate certain condensed matter phenomena such as superconductivity. In this thesis, we show how technical problems of optical lattices such as restricted decoherence times, or fundamental shortcomings such as the lack of phonons or strong spin interactions, can be overcome by using current or near-future experimental techniques. We introduce a scheme that makes it possible to simulate model Hamiltonians known from high-temperature superconductivity. For this purpose, previous simulation schemes to realise the spin interaction terms are extended. We especially overcome the condition of a filling factor of exactly one, which otherwise would restrict the phase of the simulated system to a Mott-insulator. This scheme makes a large range of parameters accessible, which is difficult to cover with a condensed matter setup. We also investigate the properties of optical lattices submerged into a Bose-Einstein condensate (BEC). A weak-coupling expansion in the BEC-impurity interaction strength is used to derive a model that describes the lattice atoms in terms of polarons, i.e.~atoms dressed by Bogoliubov phonons. This is analogous to the description of electrons in solids, and we observe similar effects such as a crossover from coherent to incoherent transport for increasing temperatures. Moreover, the condensate mediates an attractive off-site interaction, which leads to macroscopic clusters at experimentally realistic parameters. Since the atoms in the lattice can also be used as a quantum register with the BEC mediating a two-qubit gate, we derive a quantum master equation to examine the coherence properties of the atomic qubits. We show that the system exhibits sub- and superdecoherence and that a fast implementation of the two-qubit gate competes with dephasing. Finally, we show how to realise the encoding of qubits in a decoherence-free subspace (DFS) using optical lattices. We develop methods for implementing robust gate operations on qubits encoded in a DFS exploiting collisional interactions between the atoms. We also give a detailed analysis of the performance and stability of the gate operations and show that a robust implementation of quantum repeaters can be achieved using our setup. We compare the robust repeater scheme to one that makes use of conventional qubits only, and show the conditions under which one outperforms the other.

Les simulateurs quantiques, qui utilisent un système quantique contrôlable pour étudier le comportement et les propriétés d'un autre système quantique, moins accessible, sont une ressource prometteuse. Dans les dernières années, des progrès significatifs ont été faits dans de nombreux domaines expérimentaux et théoriques. Les marches quantiques à temps discret sont des systèmes simples et sophistiqués. En particulier, il a été montré qu'à la limite continue, ces marches peuvent simuler certaines théories de champs. Dans ce travail de thèse, lesdites marches sont utilisées pour explorer certains sujets d'intérêt physique, qui s'articulent autour de trois axes : (i) la connexion entre les propriétés géométriques de la marche et celles de divers champs de jauge ; (ii) la limite classique et la limite quasi-quantique, en relation surtout avec les théories de champs ; (iii) l'équilibration spontanée pour certains modèles non linéaires de marches quantiques. Chaque résultat est appuyé par une étude numérique et analytique. / Problems too demanding for classical computers can be approached promisingly with quantum simulators, which operate using one controllable quantum system in order to investigate the behavior and properties of a less accessible one. Over the past few years, significant progress has been made in a number of experimental and theoretical fields. Quantum Walks (QWs) are simple and sophisticated discrete space and time dynamical systems and it has been shown that in the continuous limit different emergent quantum fields can be simulated. In this thesis we will draw on QWs to further explore various areas of interest in Physics. More specifically our analysis will branch out into three main directions: (i) the connection between QWs and quantum field theory, with particular attention to bridging the quantum coin of QWs with the geometrical properties of gauge field theories; (ii) the study of QWs' classical limit and of the transient semi-classical dynamics, especially in relation with field theories; (iii) the spontaneous equilibration and thermalization in some nonlinear QWs-like models. Every step of this thesis will be validated by specific analytical results and numerical implementations.

Abstract Iris The power of quantum computing lies in its ability to perform certain calculations and solve complex problems exponentially faster than classical computers. This potential has profound implications for a wide range of fields, including particle physics. This thesis lays a fundamental foundation for understanding quantum computing. Particular emphasis is placed on the intricate process of quantum gate decomposition, an elementary lynchpin that underpins the development of quantum algorithms and plays a crucial role in this research. In particular, this concerns the implementation of quantum algorithms designed to simulate the dynamic evolution of multi-particle quantum systems - so-called Hamiltonian simulations. The concept of quantum gate decomposition is introduced and linked to quantum circuit optimisation. The decomposition of quantum gates plays a crucial role in fault-tolerant quantum computing in the sense that an optimal implementation of a quantum gate is essential to efficiently perform a quantum simulation, especially for near-term quantum computers. Part of this thesis aims to propose a new explicit tensorial notation of quantum computing. Two notations are commonly used in the literature. The first is the Dirac notation and the other standard formalism is based on the so-called computational basis. The main disadvantage of the latter is the exponential growth of vector and matrix dimensions and the fact that it hides some relevant quantum properties of the operations by increasing the apparent number of independent variables. A third possible notation is introduced here, which describes qubit states as tensors and quantum gates as multilinear or quasi-multilinear maps. Some advantages for the detection of separable and entangled systems and for measurement techniques are also shown. Finally, this thesis demonstrates the advantage of quantum computing in the description of multi-particle quantum systems by proposing a quantum algorithm to simulate collective neutrino oscillations. Collective flavour oscillations of neutrinos due to forward neutrino-neutrino scattering provide an intriguing many-body system for time evolution simulations on a quantum computer. These phenomena are of particular interest in extreme astrophysical settings such as core-collapse supernovae, neutron star mergers and the early universe. A detailed description of the physical phenomena and environments in which collective flavor oscillations occur is first reported, and the derivation of the Hamiltonian governing the evolution of flavor oscillations is detailed. The aim is to reproduce this evolution using a quantum algorithm. To manage the computational complexity, we use the Trotter approximation of the time evolution operator, which mitigates the exponential growth of circuit complexity. The quantum algorithm was designed to work on a trapped-ion based testbed (the theory of which is presented in detail). After machine-aware optimisation, the quantum circuit implementing the algorithm was run on the real quantum machine 'Quantinuum', and the results are presented and discussed.

L'hydrotraitement est un procédé catalytique important dans le raffinage du pétrole qui utilise des catalyseurs bimétalliques sulfurés NiWS ou NiMoS (ou CoMoS) supportés sur alumine. Leur mode conventionnel de préparation implique l’imprégnation d'une solution aqueuse de sels de Mo/W et de Ni/Co, puis l’activation par un agent sulfo-réducteur (H2S/H2). Pour répondre aux exigences environnementales et améliorer l'efficacité de l'hydrotraitement, des améliorations permanentes de la performance de ces systèmes catalytiques sont attendues. Ce travail se concentre sur la préparation de catalyseurs d'hydrotraitement hautement actifs par une approche de chimie de surface contrôlée (CSC) qui implique l'imprégnation successive de précurseurs moléculaires de MoV et NiII en solvant organique sur un support silice-alumine traité thermiquement. Dans la première partie de cette thèse, la genèse de la phase active du catalyseur CSC et conventionnel Mo et NiMo est étudiée par quick-XAS combinée à d’autres techniques (chimiométrie, XPS, RPE, STEM-HAADF, modélisation moléculaire). Nous proposons ainsi des structures moléculaires depuis les précurseurs oxydes de Mo et Ni supportés jusqu’aux nombreuses espèces intermédiaires (oxysulfure et sulfures) en fonction de la température. Cette analyse multi-technique permet d'abord de révéler les spécificités de la genèse des catalyseurs CSC et conventionnels qui peuvent expliquer leurs différentes activités catalytiques. Ensuite, elle révèle également de nouvelles connaissances sur les mécanismes d’insertion du Ni dans la phase NiMoS en fonction de la préparation. Dans la seconde partie, la possibilité de remplacer Co et Ni comme promoteurs est explorée. Ceci est entrepris en synthétisant des catalyseurs alternatifs de type XYMoS, où X et Y sont des métaux de transition 3d. Comme suggéré par des études de modélisation quantiques antérieures, certaines formulations XYMoS peuvent présenter un effet de synergie analogue à ceux des phases actives CoMoS et NiMoS. L’étude des formulations les plus prometteuses méritent d'être approfondies afin de mieux comprendre leur fonctionnement. / Hydrotreating is an important catalytic process in petroleum refining which uses sulfided bimetallic catalysts NiWS or NiMoS (or CoMoS) supported on alumina. Their conventional preparation involves an incipient wetness impregnation of an aqueous solution of Mo/W and Ni/Co salts, and then activation by a sulfo-reductive agent (such as H2S/H2). To meet environmental regulations and improve the energy efficiency of hydrotreatment, permanent improvements on the performance of these catalytic systems are expected. This work is thus focused on the preparation of highly active hydrotreating catalysts through a controlled surface chemistry (CSC) approach; which involves the successive impregnation of Mo5+ and Ni2+ molecular precursors in an organic solvent on a thermally treated silica-alumina support. In the first part of this thesis, the active phase genesis of CSC and conventional Mo and NiMo catalysts is studied by in situ quick-XAS combined with various other techniques (chemometrics, XPS, EPR, STEM-HAADF, molecular modeling). We thus propose molecular structures from the oxide of supported Mo and Ni precursors up to the numerous intermediate sulfided species as a function of temperature. This multi-technique analysis enables first to reveal the specific features of the genesis of CSC and conventional catalysts which may explain their different catalytic activities. Then, it also reveals new insights into the mechanisms of Ni promoter incorporation into the NiMoS phase as a function of the preparation. In the second part, the feasibility of replacing Co and Ni as promoters is explored. Using the CSC method, we attempted to synthesize alternative catalysts of the form XYMoS ternary sulfides, where X and Y are 3d transition metals. As suggested by previous quantum simulations, certain XY formulations possibly reveal a synergy effect as observed in CoMoS and NiMoS active phases. The most promising formulations merit further investigations.

Quantum simulations, where controllable quantum systems are used to simulate other quantum systems, originally proposed by Richard Feynman, are one of the most remarkable applications of quantum information science. Compared to computation, quantum simulations require much less number of qubits for the m to be practical. In the work described in this thesis, we have performed a few quantum simulations of small quantum systems using Nuclear Magnetic Resonance(NMR) techniques. These simulations have been used to experimentally demonstrate the underlying interesting quantum protocols. All the experiments presented have been carried out using liquid-state or liquid crystal NMR. Numerical pulse optimization techniques have been utilized in some of the experiments, to achieve better control over the spin systems. The first chapter contains “Introduction” to quantum information processing, NMR, and numerical pulse optimization techniques. In chapter 2, we describe quantum simulation of a 3-spin Heisenberg-XY spin chain having only nearest neighbour interactions. Recently, spin chains having pre-engineered short-range interactions have been proposed to efficiently transfer quantum information between different parts of a quantum information processor. Other important proposals involving these spin chains include generating entangled states and universal quantum computation. However, such engineered interactions do not occur naturally in any system. In such a scenario, the experimental viability of these proposals can be tested by simulating the spin chains in other controllable quantum systems. In this work, we first theoretically study the time evolution of bipartite and tripartite entanglement measures for a 3-spin open ended XY spin chain. Then, by simulating the XY interactions in a 3-spin nuclear spin system, we experimentally generate, (i)a bipartite maximally(pseudo-)entangled state(Bell state) between end qubits, and(ii) multipartite(pseudo-)entangled states(Wand GHZ states),starting from separable pseudo-pure states. Bell state has been generated by using only the natural unitary evolution of the XY spin chain. W-state and GHZ-state have been generated by applying a single-qubit rotation to the second qubit, and a global rotation of all the three qubits respectively after the unitary evolution of the spin chain. In chapter 3, we simulate a 3-spin quantum transverse Ising spin system in a triangular configuration, and show that multipartite quantum correlations can be used to distinguish between the frustrated and non-frustrated regimes in the ground state of this spin system. The ground state of the spin system has been prepared by using adiabatic state preparation method. Gradient ascent pulse engineering technique has been utilized to efficiently realize the adiabatic evolution of the spin system. To analyse the experimental ground state of the system, we employ two different multipartite quantum correlation measures, generated from monogamy studies of bipartite quantum correlations. Chapter 4 contains a digital quantum simulation of the mirror inversion propagator corresponding to the time evolution of an XY spin chain. This simulation has been used to experimentally demonstrate the mirror inversion of quantum states, proposed by Albanese et al.[Phys.Rev.Lett.93,230502(2004)], by which entangled states can be transferred from one end of the chain to the other end. The experiments have been performed in a 5-qubit dipolar coupled nuclear spin system. For simulation, we make use of the recently proposed unitary operator decomposition algorithm along with the numerical pulse optimization techniques, which assisted in achieving high experimental fidelities. Chapter 5 contains a digital quantum simulation of the unitary propagator of a transverse Ising spin chain, which has been used to experimentally demonstrate the perfect state transfer protocol of Di Franco et al. [Phys.Rev.Lett.101,230502(2008)]. The importance of this protocol arises due to the fact that it achieves perfect state transfer from one end of the chain to the other end without the necessity of initializing the intermediate spins of the chain, whereas most of the previously proposed protocols require initialization. The experiments have been performed in a 3-spin nuclear spin system. The simulation has also been used to demonstrate the generation of a GHZ state.