  Learning Hamiltonian of a quantum system is indispensable for prediction of
the system dynamics and realization of high fidelity quantum gates. However, it
is a significant challenge to efficiently characterize the Hamiltonian when its
Hilbert space dimension grows exponentially with the system size. Here, we
experimentally demonstrate an adaptive method to learn the effective
Hamiltonian of an 11-qubit quantum system consisting of one electron spin and
ten nuclear spins associated with a single Nitrogen-Vacancy center in a
diamond. We validate the estimated Hamiltonian by designing universal quantum
gates based on the learnt Hamiltonian parameters and demonstrate high-fidelity
gates in experiment. Our experimental demonstration shows a well-characterized
11-qubit quantum spin register with the ability to test quantum algorithms and
to act as a multi-qubit single node in a quantum network.
