Beyond qubits: Quantum optimization and lattice gauge theories in a qudit framework

Quantum computing holds the promise of tackling problems that are intractable for classical machines, yet the path from current noisy intermediate-scale quantum (NISQ) devices to fault-tolerant processors demands both better algorithms and a deeper understanding of how to exploit the full structure of quantum hardware. A key direction in this effort is the move beyond qubits: exploiting higher-dimensional local Hilbert spaces — qudits — to encode information, enforce constraints, and simulate complex physical systems more naturally and efficiently. This thesis investigates two interconnected research fronts in which the qudit framework provides concrete benefits: quantum optimization and the quantum simulation of lattice gauge theories (LGTs). On the optimization side, we show how a single ancilla qudit can enforce inequality constraints in the Quantum Approximate Optimization Algorithm (QAOA) without inflating the solution space, and how counterdiabatic driving, enhanced by symmetry reduction, improves variational performance for qudit-based algorithms. We additionally study the noise robustness of variational algorithms on real hardware, and investigate how nonstabilizerness builds up and relates to optimization performance across both qubit and qutrit protocols. On the simulation side, we analyze the ground-state structure and phase transitions of non-Abelian lattice gauge theories with dihedral symmetry, characterizing confinement and string breaking in DN models, and probing the quantum complexity of ground states across ZN , D3, and SU(2) gauge theories. Together, these results clarify where and how the qudit framework provides algorithmic and representational advantages over the standard qubit paradigm, and contribute to the broader effort of developing practical quantum simulation and optimization tools for near- term hardware