  Nuclear spins were among the first physical platforms to be considered for
quantum information processing, because of their exceptional quantum coherence
and atomic-scale footprint. However, their full potential for quantum computing
has not yet been realized, due to the lack of methods to link nuclear qubits
within a scalable device combined with multi-qubit operations with sufficient
fidelity to sustain fault-tolerant quantum computation. Here we demonstrate
universal quantum logic operations using a pair of ion-implanted 31P donor
nuclei in a silicon nanoelectronic device. A nuclear two-qubit controlled-Z
gate is obtained by imparting a geometric phase to a shared electron spin, and
used to prepare entangled Bell states with fidelities up to 94.2(2.7)%. The
quantum operations are precisely characterised using gate set tomography (GST),
yielding one-qubit average gate fidelities up to 99.95(2)%, two-qubit average
gate fidelity of 99.37(11)% and two-qubit preparation/measurement fidelities of
98.95(4)%. These three metrics indicate that nuclear spins in silicon are
approaching the performance demanded in fault-tolerant quantum processors. We
then demonstrate entanglement between the two nuclei and the shared electron by
producing a Greenberger-Horne-Zeilinger three-qubit state with 92.5(1.0)%
fidelity. Since electron spin qubits in semiconductors can be further coupled
to other electrons or physically shuttled across different locations, these
results establish a viable route for scalable quantum information processing
using donor nuclear and electron spins.
