  Protecting the dynamics of coupled quantum systems from decoherence by the
environment is a key challenge for solid-state quantum information processing.
An idle qubit can be efficiently insulated from the outside world via dynamical
decoupling, as has recently been demonstrated for individual solid-state
qubits. However, protection of qubit coherence during a multi-qubit gate poses
a non-trivial problem: in general the decoupling disrupts the inter-qubit
dynamics, and hence conflicts with gate operation. This problem is particularly
salient for hybrid systems, wherein different types of qubits evolve and
decohere at vastly different rates. Here we present the integration of
dynamical decoupling into quantum gates for a paradigmatic hybrid system, the
electron-nuclear spin register. Our design harnesses the internal resonance in
the coupled-spin system to resolve the conflict between gate operation and
decoupling. We experimentally demonstrate these gates on a two-qubit register
in diamond operating at room temperature. Quantum tomography reveals that the
qubits involved in the gate operation are protected as accurately as idle
qubits. We further illustrate the power of our design by executing Grover's
quantum search algorithm, achieving fidelities above 90% even though the
execution time exceeds the electron spin dephasing time by two orders of
magnitude. Our results directly enable decoherence-protected interface gates
between different types of promising solid-state qubits. Ultimately, quantum
gates with integrated decoupling may enable reaching the accuracy threshold for
fault-tolerant quantum information processing with solid-state devices.
