  We propose a nonadiabatic non-Abelian geometric quantum operation scheme to
realize universal quantum computation with mesoscopic Rydberg atoms. A single
control atom entangles a mesoscopic ensemble of target atoms through long-range
interactions between Rydberg states. We demonstrate theoretically that both the
single qubit and two-qubit quantum gates can achieve high fidelities around or
above 99.9% in ideal situations. Besides, to address the experimental issue of
Rabi frequency fluctuation (Rabi error) in Rydberg atom and ensemble, we apply
the dynamical-invariant-based zero systematic-error sensitivity (ZSS) optimal
control theory to the proposed scheme. Our numerical simulations show that the
average fidelity could be 99.98% for single ensemble qubit gate and 99.94% for
two-qubit gate even when the Rabi frequency of the gate laser acquires 10%
fluctuations. We also find that the optimized scheme can also reduce errors
caused by higher-order perturbation terms in deriving the Hamiltonian of the
ensemble atoms. To address the experimental issue of decoherence error between
the ground state and Rydberg levels in Rydberg ensemble, we introduce a
dispersive coupling regime between Rydberg and ground levels, based on which
the Rydberg state is adiabatically discarded. The numerical simulation
demonstrate that the quantum gate is enhanced. By combining strong Rydberg atom
interactions, nonadiabatic geometric quantum computation, dynamical invariant
and optimal control theory together, our scheme shows a new route to construct
fast and robust quantum gates with mesoscopic atomic ensembles. Our study
contributes to the ongoing effort in developing quantum information processing
with Rydberg atoms trapped in optical lattices or tweezer arrays.
