Abstract: Accurately controlling a quantum system is a fundamental requirement in quantum information processing and the coherent manipulation of molecular systems. The ultimate goal in quantum control is to prepare a desired state with the highest fidelity allowed by the available resources and the experimental constraints. Here we experimentally implement two optimal high-fidelity control protocols using a two-level quantum system comprising Bose-Einstein condensates in optical lattices. The first is a short-cut protocol that reaches the maximum quantum-transformation speed compatible with the Heisenberg uncertainty principle. In the opposite limit, we realize the recently proposed transitionless superadiabatic protocols in which the system follows the instantaneous adiabatic ground state nearly perfectly. We demonstrate that superadiabatic protocols are extremely robust against control parameter variations, making them useful for practical applications.