  Quantum holonomic gates hold built-in resilience to local noises and provide
a promising approach for implementing fault-tolerant quantum computation. We
propose to realize high-fidelity holonomic $(N+1)$-qubit controlled gates using
Rydberg atoms confined in optical arrays or superconducting circuits. We
identify the scheme, deduce the effective multi-body Hamiltonian, and determine
the working condition of the multiqubit gate. Uniquely, the multiqubit gate is
immune to systematic errors, i.e., laser parameter fluctuations and motional
dephasing, as the $N$ control atoms largely remain in the much stable qubit
space during the operation. We show that $C_N$-NOT gates can reach same level
of fidelity at a given gate time for $N\leq5$ under a suitable choice of
parameters, and the gate tolerance against errors in systematic parameters can
be further enhanced through optimal pulse engineering. In case of Rydberg
atoms, the proposed protocol is intrinsically different from typical schemes
based on Rydberg blockade or antiblockade. Our study paves a new route to build
robust multiqubit gates with Rydberg atoms trapped in optical arrays or with
superconducting circuits. It contributes to current efforts in developing
scalable quantum computation with trapped atoms and fabricable superconducting
devices.
