Solving quantum dynamics is an exponentially difficult problem. Thus, an exact numerical solution is inaccessible for any condensed matter system. A promising approach is to divide the system into a quantum subsystem containing degrees of freedom which are of greater interest or those which have more profound quantum character (e.g., have smaller mass) and a classical bath containing the rest of the system. Imposing such a partition and treating the bath classically results in quantum-classical dynamics. The quantum-classical Liouville equation is a general equation in the Hilbert space of quantum degrees of freedom while it resides in the phase space of the classical degrees of freedom. Any numerical solution to this equation requires representation of the quantum subsystem in some basis. Solutions to this equation have been already proposed in the subsystem, adiabatic and force bases, each with its own cons and pros. In this work, the quantum-classical equations of motion are cast in the subsystem basis and subsequently mapped to a number of fictitious harmonic oscillators. The result is quantum-classical dynamics in the mapping basis which treats both quantum and classical degrees of freedom on the same footing, i.e., in phase space. Neglecting a portion of the back reaction of the quantum-subsystem to classical bath results in an expression for the time evolution of an operator (density matrix) equal to its Poisson bracket with the Hamiltonian. This equation can be solved in terms of characteristics to provide a computationally tractable method for calculating quantum-classical dynamical properties. The expressions for expectation values and correlation functions in this formalism are derived. Calculations on spin-boson system, barrier crossing models---the so called Tully models---and the Fenna-Mathews-Olson pigments show very good agreement between the results of this method and numerical solutions to the Schrödinger equation.

Solving quantum dynamics is an exponentially difficult problem. Thus, an exact numerical solution is inaccessible for any condensed matter system. A promising approach is to divide the system into a quantum subsystem containing degrees of freedom which are of greater interest or those which have more profound quantum character (e.g., have smaller mass) and a classical bath containing the rest of the system. Imposing such a partition and treating the bath classically results in quantum-classical dynamics. The quantum-classical Liouville equation is a general equation in the Hilbert space of quantum degrees of freedom while it resides in the phase space of the classical degrees of freedom. Any numerical solution to this equation requires representation of the quantum subsystem in some basis. Solutions to this equation have been already proposed in the subsystem, adiabatic and force bases, each with its own cons and pros. In this work, the quantum-classical equations of motion are cast in the subsystem basis and subsequently mapped to a number of fictitious harmonic oscillators. The result is quantum-classical dynamics in the mapping basis which treats both quantum and classical degrees of freedom on the same footing, i.e., in phase space. Neglecting a portion of the back reaction of the quantum-subsystem to classical bath results in an expression for the time evolution of an operator (density matrix) equal to its Poisson bracket with the Hamiltonian. This equation can be solved in terms of characteristics to provide a computationally tractable method for calculating quantum-classical dynamical properties. The expressions for expectation values and correlation functions in this formalism are derived. Calculations on spin-boson system, barrier crossing models---the so called Tully models---and the Fenna-Mathews-Olson pigments show very good agreement between the results of this method and numerical solutions to the Schrödinger equation.

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Quantum computers have seen rapid development over the last two decades. Despite this, they are not yet scalable or fault-tolerant (i.e. we cannot address arbitrarily many error-corrected qubits). Therefore, improvements that include consideration of the underlying physics are paramount. To do this, we must reduce existing errors and understand how algorithms perform without error correction. In this dissertation, we provide contributions toward these goals. We organize these efforts into three groups. Firstly, we focus on quantum control. We introduce a novel scheme for performing entangling gates on superconducting qubits. We create fast, high-fidelity entangling operations and single-qubit gates to implement arbitrary quantum operations. Then, we implement entangling gates on real transmon qubits. Finally, we develop new techniques for entangling gates on spin qubits. In total, we improve low-level device performance with high-fidelity entangling operations. Secondly, we focus on quantum simulation algorithms. First, we apply error mitigation techniques to a quantum simulation algorithm while simultaneously performing device characterization. Then we take advantage of known symmetries of the input Hamiltonian to improve the same algorithm. Then, we demonstrate that this reduces resources compared to other approaches in the presence of noise. Then we compare this technique with state-of-the-art approaches. Then, we improve this algorithm with approaches from quantum control. Finally, we develop a novel algorithm to simulate spin chains on a quantum processor with improved resources compared to other techniques. In total, we improve quantum simulation algorithms, with the aim of better utilizing current devices. Thirdly, we consider the ADAPT-VQE algorithm, which is used to construct quantum circuits for preparing trial states in quantum simulation. In total, we improve gate counts for the algorithm, improve a separate algorithm by utilizing the gradient criterion, and leverage the repeating structure of an input Hamiltonian to improve performance. Finally, we provide a deeper understanding of ADAPT-VQE and demonstrate its robustness to scaling issues of competing algorithms. In total, we improve the algorithm and its applicability. Thus, we improve quantum simulation algorithms that can be run in the near term. / Doctor of Philosophy / The computers that we interact with every day rely on the processing of bits, represented as 1's or 0's. The rules that govern how they operate mostly rely on classical physics (i.e. discovered before quantum physics), which does not include any quantum effects. If we instead allow for quantum rules and quantum bits ("qubits"'), new types of algorithms are possible. That is to say, quantum computers can do some things more efficiently than classical computers. As such, there is a massive effort to build these devices. Because these devices are so delicate and in their early stages, this requires an understanding of the algorithm and the physical device performing it. Therefore, improving the overall performance requires taking high and low-level aspects of this design into consideration. In this dissertation, we provide three groups of contributions to achieving this goal. In the first group, we improve the device performance by considering how operations are performed on qubits, primarily in terms of producing quantum operations that have no classical analog. In the second group, we improve the simulation of quantum systems on quantum devices with a focus on how existing imperfections in the device impact the results. In the third group, we make improvements to an algorithm used to simulate quantum systems like molecules, while also developing a deeper understanding of how the algorithm functions. In each of these parts, we develop novel techniques to improve device and algorithm performance, contributing to the applicability and utility of current and future quantum devices.

In this thesis quantum-classical dynamics is applied to the study of quantum condensed phase processes. This approach is based on the quantum-classical Liouville equation where the dynamics of a small subset of the degrees of freedom are treated quantum mechanically while the remaining degrees of freedom are treated by classical mechanics to a good approximation. We use this approach as it is computationally tractable, and the resulting equation of motion accurately accounts for the quantum and classical dynamics, as well as the coupling between these two components of the system. By recasting the quantum-classical Liouville equation into the form of a generalized master equation we investigate connections to surface-hopping. The link between these approaches is decoherence arising from interaction of the subsystem with the environment. We derive an evolution equation for the subsystem which contains terms accounting for the effects of the environment. One of these terms involves a memory kernel that accounts for the coherent dynamics. If this term decays rapidly, a Markovian approximation can be made. By lifting the resulting subsystem master equation into the full phase space, we obtain a Markovian master equation that prescribes surface-hopping-like dynamics. Our analysis outlines the conditions under which such a description is valid. Next, we consider the calculation of the rate constant for a quantum mechanical barrier crossing process. Starting from the reactive-flux autocorrelation function, we derive a quantum-classical expression for the rate kernel. This expression involves quantum-classical evolution of a species operator averaged over the initial quantum equilibrium structure of the system making it possible to compute the rate constant via computer simulation. Using a simple model for a proton transfer reaction we compare the results of the rate calculation obtained by quantum-classical Liouville dynamics with that of master equation dynamics. The master equation provides a good approximation to the full quantum-classical Liouville calculation for our model and a more stable algorithm results due to the elimination of oscillating phase factors in the simulation. Finally, we make use of the theoretical framework established in this thesis to analyze some aspects of decoherence used in popular surface-hopping techniques.

In this thesis quantum-classical dynamics is applied to the study of quantum condensed phase processes. This approach is based on the quantum-classical Liouville equation where the dynamics of a small subset of the degrees of freedom are treated quantum mechanically while the remaining degrees of freedom are treated by classical mechanics to a good approximation. We use this approach as it is computationally tractable, and the resulting equation of motion accurately accounts for the quantum and classical dynamics, as well as the coupling between these two components of the system. By recasting the quantum-classical Liouville equation into the form of a generalized master equation we investigate connections to surface-hopping. The link between these approaches is decoherence arising from interaction of the subsystem with the environment. We derive an evolution equation for the subsystem which contains terms accounting for the effects of the environment. One of these terms involves a memory kernel that accounts for the coherent dynamics. If this term decays rapidly, a Markovian approximation can be made. By lifting the resulting subsystem master equation into the full phase space, we obtain a Markovian master equation that prescribes surface-hopping-like dynamics. Our analysis outlines the conditions under which such a description is valid. Next, we consider the calculation of the rate constant for a quantum mechanical barrier crossing process. Starting from the reactive-flux autocorrelation function, we derive a quantum-classical expression for the rate kernel. This expression involves quantum-classical evolution of a species operator averaged over the initial quantum equilibrium structure of the system making it possible to compute the rate constant via computer simulation. Using a simple model for a proton transfer reaction we compare the results of the rate calculation obtained by quantum-classical Liouville dynamics with that of master equation dynamics. The master equation provides a good approximation to the full quantum-classical Liouville calculation for our model and a more stable algorithm results due to the elimination of oscillating phase factors in the simulation. Finally, we make use of the theoretical framework established in this thesis to analyze some aspects of decoherence used in popular surface-hopping techniques.

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Le calcul de quantités dépendants du temps pour des systèmes quantiques est limité le scaling exponentiel des méthodes exactes. Néanmoins, ces quantités présentes un intérêt scientifique important. Un compromis, entre précision et coût, est trouvé par les méthodes quasi-classiques. Dans ces méthodes, la densité thermique exacte est combinée à des trajectoires approximant la dynamique quantique. Durant ma thèse, j’ai développé et appliqué une méthode quasi-classique : PIM (Phase Integration Methode) qui combine des algorithmes MC et MD pour calculer des fonction de corrélation. Le Chapitre 2 décrit les méthodes quasi-classiques ainsi que les approximations qui permettent d’en tirer les fonctions de corrélations dépendants du temps.Le Chapitre 3 illustre comment PIM est adapté au calcul de la densité de Wigner qui est une quantité clé pour les méthodes quasi-classique. À travers le calcul de cette quantité, PIM est capable de capturer des corrélations entre différents degrés de liberté. Dans le Chapitre 4, on montre comment PIM est adapté au calcul de spectres infrarouge. La comparaison des résultats avec d’autres méthodes montre que PIM est une méthode précise pour les systèmes à basse dimensionnalité. Les spectres de OH et CH4 confirment que PIM ne souffrent pas de problèmes intrinsèques comme CMD ou RPMD et peut être appliqué à des systèmes à plus haute dimensionnalité. Le Chapitre 5 présente la méthodologie pour calculer des constantes de vitesse à l’aide de PIM. Les résultats sont bons jusqu’à 300 K mais pas en dessous. Le travail futur se concentrera sur le calcul de la fonction de corrélation de Kubo flux-side pour remédier à ce problème. / Simulation of time-dependent quantities for quantum systems is limited by the exponential scaling of exact methods. However, the calculation of these quantities is key in many problems. A reasonable compromise among accuracy and cost is done by the quasi-classical methods for computing time correlation functions. In these methods, the thermal density is combined with trajectories that approximate quantum dynamics. In my thesis, I develop and apply quasi-classical methods for vibrational spectroscopy. The focus is on the Phase Integration Method. PIM is based on combining MD and MC algorithms to compute appropriate quantities. Chapter 2 is devoted to a general description of the quasi-classical methods. We introduce the different approximations used to compute quantum time correlation functions. Chapter 3 illustrates how PIM is adapted to the calculation of the Wigner density, which is a key quantity in quasi-classical methods. Via this quantity, we show that PIM is able to capture quantum correlation effects among different degrees of freedom. Chapter 4 focuses on the adaptation of PIM for the infrared spectroscopy. Comparison of our results, show that PIM is accurate for low dimensional models. OH and CH4 spectrum confirms that our approach does not suffer from the pathologies such as CMD and RPMD but also that it can treat systems with a larger number of degrees of freedom reliably. Chapter 5 presents the methodology used to calculate rate constants with PIM. The results are in good agreement with the exact reference until 300 K. Future work will focus on using the Kubo flux side correlation function.

Объект исследования – линейно-цепочечный углерод на подложке кремния. Цель работы – моделирование синтеза цепочечного углерода методом гибридной молекулярной динамики. Методы исследования: гибридная квантово-классическая молекулярная динамика. В ходе выполнения данной магистерской диссертации на языке Python были написаны программы, запускающие моделирование квантово-классической молекулярной динамики. В результате был разработан модуль для реализации электростатического подхода QM/MM вычислений. На основе данного модуля было произведено моделирование ионно-плазменного синтеза углеродной пленки. Анализа полученных структур показал, что содержание sp1, sp2, sp3 углерода в пленке максимально при энергиях аргона 40, 70, 90 эВ соответственно. / The object under study is linear-chained carbon on a silicon substrate. The aim of this work is linear-chained carbon synthesis simulation by the hybrid molecular dynamics. Research methods: hybrid quantum-classical molecular dynamics. In the course of this master's dissertation, Python programs were written. That programs run the simulation of quantum-classical molecular dynamics. As a result, a module was developed to implement the electrostatic approach of QM/MM calculations. On the basis of this module, modeling of the ion-beam plasma synthesis of a carbon film was carried out. The obtained structures analysis showed that the content of sp1, sp2, and sp3 carbon in the film is maximum at argon energies of 40, 70, and 90 eV, respectively.

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