Quantum networks require functional nodes consisting of stationary registers with the capability of high-fidelity quantum processing and storage, which efficiently interface with photons propagating in an optical fiber. We report a significant step towards realization of such nodes using a diamond nanocavity with an embedded silicon-vacancy (SiV) color center and a proximal nuclear spin. Specifically, we show that efficient SiV-cavity coupling (with cooperativity $C >30$) provides a nearly-deterministic interface between photons and the electron spin memory, featuring coherence times exceeding one millisecond. Employing coherent microwave control, we demonstrate heralded single photon storage in the long-lived spin memory as well as a universal control over a cavity-coupled two-qubit register consisting of a SiV and a proximal $^{mathrm{13}}$C nuclear spin with nearly second-long coherence time, laying the groundwork for implementing quantum repeaters.
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