  Measurement-based quantum computation relies on single qubit measurements of
large multipartite entangled states, so-called lattice-graph or cluster states.
Graph states are also an important resource for quantum communication, where
tree cluster states are a key resource for one-way quantum repeaters. A
photonic realization of this kind of state would inherit many of the benefits
of photonic platforms, such as very little dephasing due to weak environmental
interactions and the well-developed infrastructure to route and measure
photonic qubits. In this work, a linear cluster state and GHZ state generation
scheme is developed for group-IV color centers. In particular, this article
focuses on an in-depth investigation of the required control operations,
including the coherent spin and excitation gates. We choose an off-resonant
Raman scheme for the spin gates, which can be much faster than microwave
control. We do not rely on a reduced level scheme and use efficient
approximations to design high-fidelity Raman gates. We benchmark the
spin-control and excitation scheme using the tin vacancy color center coupled
to a cavity, assuming a realistic experimental setting. Additionally, the
article investigates the fidelities of the Raman and excitation gates in the
presence of radiative and non-radiative decay mechanisms. Finally, a quality
measure is devised, which emphasizes the importance of fast and high-fidelity
spin gates in the creation of large entangled photonic states.
