Comparing the performance of practical two-qubit gates for individual $^{171}$Yb ions in yttrium orthovanadate

In this paper, we investigate three schemes for implementing Controlled-Z (CZ) gates between individual ytterbium (Yb) rare-earth ions doped into yttrium orthovanadate (YVO$_4$ or YVO). Specifically, we investigate the CZ gates based on magnetic dipolar interactions between Yb ions, photon scattering off a cavity, and a photon interference-based protocol, with and without an optical cavity. We introduce a theoretical framework for precise computations of state and gate infidelities, accounting for noise effects. We then compute the state fidelity for each scheme to evaluate the feasibility of their experimental implementation. Based on these results, we compare the performance of the two-qubit gate schemes and discuss their respective advantages and disadvantages. We conclude that the probabilistic photon interference-based scheme offers the best fidelity scaling with cooperativity and is superior with the current technology of Yb values, while photon scattering is nearly deterministic but slower with less favourable fidelity scaling as a function of cooperativity. The cavityless magnetic dipolar scheme provides a fast, deterministic gate with decent fidelities if close ion localization can be realized. While focusing on $^{171}$Yb$^{3+}$:YVO system as a case study, the theoretical tools and approaches developed in this work are broadly applicable to other spin qubit systems.