Optimal control of two-level quantum systems based on the geometrical parametrization

This paper presents an approach based on constructing control Hamiltonians to steer qubit states. When three tunable control Hamiltonians are available, we can adjust the controls to transform two-level quantum systems from an arbitrary initial pure state to another arbitrary target pure state along the unique geodesic curve on Block sphere. In terms of a kind of time-energy performance J = ∫₀^tf [λ + (1 - λ)E(u(t))]dt, we discuss how to design control magnitude to minimize this performance. An example indicates the feasibility and efficiency of this approach on optimal control of two-level quantum systems.