  Photonic cluster states are a powerful resource for measurement-based quantum
computing and loss-tolerant quantum communication. Proposals to generate
multi-dimensional lattice cluster states have identified coupled spin-photon
interfaces, spin-ancilla systems, and optical feedback mechanisms as potential
schemes. Following these, we propose the generation of multi-dimensional
lattice cluster states using a single, efficient spin-photon interface coupled
strongly to a nuclear register. Our scheme makes use of the contact hyperfine
interaction to enable universal quantum gates between the interface spin and a
local nuclear register and funnels the resulting entanglement to photons via
the spin-photon interface. Among several quantum emitters, we identify the
silicon-29 vacancy centre in diamond, coupled to a nanophotonic structure, as
possessing the right combination of optical quality and spin coherence for this
scheme. We show numerically that using this system a 2x5-sized cluster state
with a lower-bound fidelity of 0.5 and repetition rate of 65 kHz is achievable
under currently realised experimental performances and with feasible technical
overhead. Realistic gate improvements put 100-photon cluster states within
experimental reach.
