Robust and effective dynamics for quantum control

<p dir="ltr">Scalable and practical quantum technologies, such as quantum computation and quantum metrology, require both precise control of quantum systems and low error rates. However, quantum systems are highly susceptible to noise and perturbations, which can lead to inaccuracies in their dynamics. Mitigating such errors demands not only precision engineering but also a fundamental understanding supported by analytical and technical frameworks.</p><p dir="ltr">Common sources of error include incomplete knowledge of Hamiltonian parameters, fluctuations in control amplitudes, and approximation techniques that fail to accurately capture the dynamics of complex or time-dependent systems. These challenges can be overcome through the development of quantum control strategies that are robust against uncertainty, efficient implementation of control protocols, and effective modeling supported by systematic error characterization.</p><p dir="ltr">This thesis presents advancements aimed at addressing these challenges. We first develop a robust quantum control framework based on mathematical and optimal control techniques to handle continuous parameter uncertainty and analyze differences in controllability. We further develop efficient applications of optimal control to accelerate gate implementation in topological quantum computing, achieving significant speedups while maintaining high accuracy.</p><p dir="ltr">Later on, we establish a general framework that combines quantum control strategies with effective Hamiltonians to provide tighter bounds in quantum metrology by mitigating noise and harnessing quantum advantage. Lastly, we develop a non-perturbative framework for generating effective Hamiltonians of arbitrary order in periodically driven systems. This method is capable of providing error bounds for the dynamics generated by the effective and original Hamiltonians.</p><p dir="ltr">Together, these results contribute to the development of more reliable control and modeling techniques in quantum systems, with potential applications across quantum computation and quantum metrology.</p>