Quantum Annealing Complexity and Stochastic Quantum Control

Abstract

This thesis investigates two central challenges for quantum computing: identifying when quantum annealing can offer a computational advantage, and developing effective methods to control realistic quantum systems. It establishes complexity bounds for an adiabatic algorithm for combinatorial optimization, showing a universal quadratic advantage while highlighting the practical difficulty of realizing the required schedule. It also introduces Path Integral Quantum Control, a trajectory-based framework for optimizing controls in open quantum systems, and develops a stochastic Hamiltonian description of continuous quantum measurement connected to double-bracket gradient flows. Together, these results advance the theoretical foundations of quantum optimization, open-system control, and measurement-based feedback