Leakage Suppression for Holonomic Quantum Gates.

Non-Abelian geometric phases acquired in cyclic quantum evolutions can be utilized as natural resources for constructing robust holonomic gates for quantum information processing. Recently, an extensible holonomic quantum computation (HQC) was proposed and demonstrated in a recent superconducting experiment [T. Yan et al., Phys. Rev. Lett. 122, 080501 (2019)]. However, for the weakly anharmonic system, this HQC was given of low gate fidelity due to leakage to states outside of the computational subspace. Here, we propose a scheme that nonadiabatic holonomic gates can be constructed via dynamical invariant using resonant superconducting interaction of three-level quantum systems. Furthermore, we can be compatible with optimal control technology for maximizing the gate fidelity against leakage error. For benchmarking, we provide a thorough analysis on the performance of our scheme under experimental conditions; we found that both the gate error can be reduced by as much as 91.7\% compared with conventional HQC. Furthermore, the leakage rates can be reduced to $10^{-4}$ level by numerically choosing suitable control parameter. Therefore, our scheme provides a promising way towards fault-tolerant quantum computation in a weakly anharmonic solid-state system.