Robust paths to realize nonadiabatic holonomic gates

To realize one desired nonadiabatic holonomic gate, various equivalent evolution paths can be chosen. However, in the presence of errors, these paths become inequivalent. In this paper, we investigate the difference of these evolution paths in the presence of systematic Rabi frequency errors and aim to find paths with optimal robustness to realize one-qubit nonadiabatic holonomic gates. We focus on three types of evolution paths in the $\Lambda$ system: paths belonging to the original two-loop scheme [New J. Phys. {\bf 14}, 103035 (2012)], the single-loop multiple-pulse scheme [Phys. Rev. A {\bf 94}, 052310 (2016)], and the off-resonant single-shot scheme [Phys. Rev. A {\bf 92}, 052302 (2015); Phys. Lett. A {\bf 380}, 65 (2016)]. Whereas both the single-loop multiple-pulse and single-shot schemes aim to improve the robustness of the original two-loop scheme by shortening the exposure to decoherence, we here find that the two-loop scheme is more robust to systematic errors in the Rabi frequencies. More importantly, we derive conditions under which the resilience to this kind of error can be optimized, thereby strengthening the robustness of nonadiabatic holonomic gates.