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I developed a computationally efficient framework for accelerating the quantum optimal control of various multi-qubit systems. This framework decomposes the Hilbert space of the multi-qubit system and enables unitary transformations of the Hamiltonians based on the symmetry of finite groups. The Hamiltonians are block diagonalized after transformation, which features a natural structure for computing these blocks in parallel. Specifically, the size of the Hamiltonians of an n-qubit system is reduced from 2^n × 2^n to O(n × n) or O((2^n / n) × (2^n / n)) under Sn symmetry or Dn symmetry, respectively. This approach reduces the execution time of quantum optimal control by orders of magnitude while the accuracy of the output is not affected. The Lie-Trotter-Suzuki decomposition generalizes this symmetry-based approach to a more general variety of multi-qubit systems. Based on the symmetry-induced decomposition of the Hilbert space, I propose the concept of symmetry-protected subspaces, which are potential platforms for preparing commonly used symmetric states, realizing simultaneous gate operations, quantum error suppression, and simulation of other quantum systems. A perspective on ladder operators and selection rules is provided to facilitate the understanding of the transformation of the Hamiltonians. I provide the Python source code for the quantum optimal control framework and the symmetry-based methods.

Density functional theory (DFT) is an effective computational model, which enables calculations of properties and dynamical evolution under external fields for quantum many-body systems from first principles. On the other hand, there has been a burgeoning interest in addressing the ``inverse" problem: can we design a control field to steer a quantum system toward a desired configuration? Quantum Optimal Control (QOC) has risen to prominence as a potent framework in this regard. A lot of progress has been made, especially for the finite-dimensional quantum spin system. However, the optimal control of interacting many-body systems is a relatively young research field.

The first part of this dissertation presents a computational scheme that integrates the exact nonlocal exchange operator into ground-state calculations for multi-shell nanowires with various cross-sectional shapes, employing the finite element method. This method is applied to several core-shell nanowires, underscoring the crucial role of the nonlocal exchange operator. We demonstrate its significant influence on electronic properties, such as electron occupancy numbers, energy eigenvalues, energy separations, and electron localization patterns.

The latter half of this work delineates a computational methodology for applying QOC to interacting many-body systems within arbitrary geometric domains within the DFT context. Employing the Lagrangian multiplier method, we derive the gradient expression for the loss functional. A propagator integration method (Green's function) is implemented to evolve wavefunctions forward and backward, incorporating the WKB approximation to accommodate spatially varying effective electron mass. This optimization problem is iteratively solved to determine the optimal control field. Our approach is validated through a test example and subsequently applied to two complex systems, demonstrating its reliability and efficacy. These applications also allow us to investigate the effects of varying propagation times on control strategies and explore the feasibility of manipulating entire systems using localized control potentials.

The real-time time-dependent density functional theory (rt-TDDFT) approach, which is complementary to the more traditional linear-response TDDFT (lr-TDDFT), propagates the electron density in real time for studying the ground and excited states. Researchers use rt-TDDFT to study electron dynamics in real time. Moreover, rt-TDDFT can treat nonlinear effects, which is especially useful for experimentalists studying external field effects in complex systems. This dissertation presents computational methods for studying a complex system's ground and excited states. We start with ground state calculations, via density functional theory (DFT), for studies on Cyclodextrins as a catalyst, binary compound convex hull, and transition states. Then we go into ground state studies with nuclear motion using the Born-Oppenheimer Molecular Dynamics. An application of lr-TDDFT on TiSe$_{2}$ follows this. We transition to excited state studies by introducing our rt-TDDFT formalism. We validate the implementation with benchmark tests for essential elements. Finally, we demonstrate our implementation capabilities and apply them to practical systems. The first application simulates the attosecond transient absorption spectroscopy for charge transfer and polarization switching in BaTiO$_{3}$. The second application involves molecular dynamics, a nonlinear process, for photo-induced degradation mechanisms of perfluorooctanoic acid (PFOA). By explicitly accounting for non-adiabatic excited-state interactions in solvated PFOA, we show that these photo-induced excitations enable a charge-transfer process that polarizes the C\textendash F bond, resulting in a dynamic dissociation on a femtosecond time scale. Ultimately, this dissertation emphasizes the importance of quantum simulations for studying excited states of molecular and extended systems.

The deleterious impact of erosion due to high-velocity particle impingement adversely affects a variety of engineering/industrial systems, resulting in irreversible mechanical wear of materials/components. Brute force computational fluid dynamics (CFD) calculations are commonly used to predict surface erosion by directly solving the Navier Stokes equations for the fluid and particle dynamics; however, these numerical approaches often require significant computational resources, limiting the scope of observations. Furthermore, when different initial conditions are needed to analyze the system, the whole procedure of CFD calculation should be restarted de novo without recourse to previously converged cases.In contrast, recent data-driven approaches using machine learning (ML) have shown immense promise for more efficient and accurate predictions to sidestep the computationally demanding CFD calculations. This thesis proposes an ML approach using CFD data to predict erosion on a complex boiler header of an industrial coal plant. I developed a hybrid ML approach to predict particle trajectory and surface erosion rate based on initial particle parameters and positions in the OP-650 industrial boiler header. Specifically, I integrated the time-series models, such as LSTM and GPT-2, with the CNN model to predict the surface erosion rate based on the five initial parameters only. The hybrid ML architecture uses the predicted trajectories from the time-series models as input data for a CNN model to predict the surface erosion rate.

The treatment of atomic anions with Kohn–Sham density functional theory (DFT) has long been controversial because the highest occupied molecular orbital (HOMO) energy, EHOMO, is often calculated to be positive with most approximate density functionals. In Chapter 1, we assess the accuracy of orbital energies and electron affinities for all three rows of elements in the periodic table (H–Ar) using a variety of theoretical approaches and customized basis sets. Among all of the theoretical methods studied here, we find that a nonempirically tuned range-separated approach provides the best accuracy for a variety of basis sets, even for small basis sets where most functionals typically fail. While previous approaches utilize non-self-consistent methods, the nonempirically tuned range-separated procedure used here yields well-defined electronic couplings/gradients and correct EHOMO values because both the potential and resulting electronic energy are computed self-consistently. Orbital energies and electron affinities are further analyzed in the context of the electronic energy as a function of electronic number (including fractional numbers of electrons) to provide a stringent assessment of self-interaction errors for these complex anion systems. In Chapter 2, we present a new analysis of exchange and dispersion effects for calculating halogen-bonding interactions in a wide variety of complex dimers (69 total). Contrary to previous work on these systems, we find that dispersion plays a more significant role than exact exchange in accurately calculating halogen-bonding interaction energies. In particular, we find that even if the amount of exact exchange is non-empirically tuned to satisfy known DFT constraints, we still observe an overall improvement in predicting dissociation energies when dispersion corrections are applied, in stark contrast to previous studies (J. Chem. Theory Comput. 2013, 9, 1918-1931). In addition to these new analyses, we correct several (14) inconsistencies in the "XB51" set, which is widely used in the scientific literature for developing and benchmarking various DFT methods. Together, these new analyses and revised benchmarks emphasize the importance of dispersion and provide corrected reference values that are essential for developing/parameterizing new DFT functionals specifically for complex halogen-bonding interactions.