  Spins associated to solid-state colour centers are a promising platform for
investigating quantum computation and quantum networks. Recent experiments have
demonstrated multi-qubit quantum processors, optical interconnects, and basic
quantum error correction protocols. One of the key open challenges towards
larger-scale systems is to realize high-fidelity universal quantum gates. In
this work, we design and demonstrate a complete high-fidelity gate set for the
two-qubit system formed by the electron and nuclear spin of a nitrogen-vacancy
center in diamond. We use gate set tomography (GST) to systematically optimise
the gates and demonstrate single-qubit gate fidelities of up to $99.999(1)\%$
and a two-qubit gate fidelity of $99.93(5) \%$. Our gates are designed to
decouple unwanted interactions and can be extended to other electron-nuclear
spin systems. The high fidelities demonstrated provide new opportunities
towards larger-scale quantum processing with colour-center qubits.
