Optimal Control of thermally noisy quantum gates in a multilevel system

Quantum systems are inherently sensitive to environmental noise and imperfections in external control fields, posing a significant challenge for the practical implementation of quantum technologies. These noise sources degrade the fidelity of quantum gates, making their mitigation a key requirement for realizing reliable quantum computing. In this study, we apply Optimal Control Theory (OCT) within a thermodynamically consistent framework to design and stabilize high-fidelity quantum gates under Markovian noise. Our approach focuses on thermal relaxation and incorporates these effects into the control protocol, wherein external driving fields not only govern the system's unitary evolution but also modulate its interaction with the environment. By leveraging this interplay, we demonstrate that OCT can enable entropy-modifying processes, such as targeted cooling or heating, while maintaining high-fidelity gate performance in noisy environments. To validate our approach, we employ high-precision numerical methods for an open quantum system implementing one- or two-qubit gates embedded in a larger Hilbert space. The results showcase robust gate operation even under significant dissipative influences, offering a concrete path toward fault-tolerant quantum computation under realistic conditions.