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Photonic cluster states are a powerful resource for measurement-based quantumcomputing and loss-tolerant quantum communication. Proposals to generatemulti-dimensional lattice cluster states have identified coupled spin-photoninterfaces, spin-ancilla systems, and optical feedback mechanisms as potentialschemes. Following these, we propose the generation of multi-dimensionallattice cluster states using a single, efficient spin-photon interface coupledstrongly to a nuclear register. Our scheme makes use of the contact hyperfineinteraction to enable universal quantum gates between the interface spin and alocal nuclear register and funnels the resulting entanglement to photons viathe spin-photon interface. Among several quantum emitters, we identify thesilicon-29 vacancy centre in diamond, coupled to a nanophotonic structure, aspossessing the right combination of optical quality and spin coherence for thisscheme. We show numerically that using this system a 2x5-sized cluster statewith a lower-bound fidelity of 0.5 and repetition rate of 65 kHz is achievableunder currently realised experimental performances and with feasible technicaloverhead. Realistic gate improvements put 100-photon cluster states withinexperimental reach.

Multidimensional cluster states using a single spin-photon interface coupled strongly to an intrinsic nuclear register