This paper performs preparation and preservation of the Hadamard gate using the Lyapunov method in a non-Markovian two-level open quantum system embedded in a hybrid bath composed of bosons and fermions, for the first time. To this end, the dynamics of the gate operator corresponds to the unitary and dissipation parts are extracted and the explicit form of equations of decay rate factor in the hybrid environment is attained. Afterwards, the Lyapunov control method based on the matrix logarithm function is used to manipulate the propagator to the desired Hadamard gate and a limitation is defined on control laws in order to have control fields with upper and lower bound that is suitable from the practical point of view. Finally, the numerical simulations are illustrated for various conditions of correlation strength and memory time of baths. The simulation results show that the proposed control fields perfectly prepare the Hadamard gate and can achieve gate error less than 2×10^−7 and 4×10−^10 considering infinite and finite bounds on the control fields, respectively, with strong coupling between the environment and the system. Moreover, when control is bounded, the fidelity of values very close to 1 can be reached under specific conditions, and maximum fidelity in all cases is more than 0.99, and besides, the issue of excessive control amplitude is tackled. Furthermore, the gate operator can be preserved during the gate processing time under the designed control fields. The proposed approach can be applied to other single-qubit gates.
