Application of Sliding Mode Control & Observer to Stabilized Two-Level Quantum Systems

State estimation and control of quantum systems present significant challenges due to the inherent noise, uncertainty, and non-linearity of quantum dynamics. In this paper, we propose the application of Sliding Mode Observers (SMOs) as a robust and efficient solution for the state estimation and control of finite dimensional quantum systems whose state is described by a density matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\rho$</tex>. We demonstrate how these observers can be used to track quantum states accurately, despite the presence of measurement noise and system dynamics uncertainties. Furthermore, we introduce a control strategy that leverages the state estimates to stabilize the quantum system, ensuring desirable performance even under challenging conditions. Numerical simulations are provided to illustrate the effectiveness of the proposed methodology, showcasing improvements in system stability, accuracy of state estimation, and robustness to perturbations. The results highlight the potential of sliding mode observers as a viable tool for quantum control applications, including quantum information processing, quantum computing, and other emerging quantum technologies.