Precision tomography of a three-qubit electron-nuclear quantum processor in silicon

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, the full potential of nuclear spins for quantum computing has not yet been realized, due to the lack of high-fidelity multi-qubit operations, and of pathways to distribute quantum information among nuclei within a scalable device. Here we demonstrate universal quantum logic operations using a pair of ion-implanted $^{31}$P 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 gate fidelities up to 99.93(3)% and a two-qubit gate fidelity of 99.21(14)%. The GST analysis provides unprecedented details on the nature of the errors -- including the emergence of correlated and entangling errors -- which will be crucial for developing 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 nuclear spins.