Optimal control for one- and two-qubit gates as a geodesic problem - Silva, Adonai Hilario da (2025)
The main goal of this work is to make use of sub-Riemannian geometry and Pontryagins Maximum Principle to investigate optimal control for computing one- and two-qubit quantum gates, as well as to analyze how the presence of uncontrollable interactions, such as noise and cross-talk operators, influence on the energetic cost of performing such unitary operations. Initially, we study the case of a single noisy qubit. We aim to identify sub-Riemannian geodesics that connect the identity to a unitary operation, sufficiently close to the ideal case, despite the presence of decoherence noise. Next, we expand the study to the case of two qubits, initially only under a time-constant cross-talk interaction, and then, under both: cross-talk interaction and decoherence noise. For all case studies, the main challenge lies in determining the unknown parameters in the geodesic equation for the desired unitary operator. We present a novel method, based on random sampling, for determining such parameters and how they can lead to high-fidelity quantum gates which are synthesized through smooth external control fields.