  Quantum optimal control can play a crucial role to realize a set of universal
quantum logic gates with error rates below the threshold required for
fault-tolerance. Open-loop quantum optimal control relies on accurate modeling
of the quantum system under control, and does not scale efficiently with system
size. These problems can be avoided in closed-loop quantum optimal control,
which utilizes feedback from the system to improve control fidelity. In this
paper, two gradient-based closed-loop quantum optimal control algorithms, the
hybrid quantum-classical approach (HQCA) described in [Phys. Rev. Lett. 118,
150503 (2017)] and the finite-difference (FD) method, are experimentally
investigated and compared to the open-loop quantum optimal control utilizing
the gradient ascent method. We employ a solid-state ensemble of coupled
electron-nuclear spins serving as a two-qubit system. Specific single-qubit and
two-qubit state preparation gates are optimized using the closed-loop and
open-loop methods. The experimental results demonstrate the implemented
closed-loop quantum control outperforms the open-loop control in our system.
Furthermore, simulations reveal that HQCA is more robust than the FD method to
gradient noise which originates from measurement noise in this experimental
setting. On the other hand, the FD method is more robust to control field
distortions coming from non-ideal hardware
