Direct Rydberg-Rydberg interactions decay rapidly with distance, limiting blockade-based gates between spatially separated neutral-atom qubits. We study a hybrid architecture in which a trapped ion mediates an additional effective interaction between two Rydberg atoms in optical tweezers. Following the ion-mediated mechanism [S. Mudli, S. Mal. S. S. Rej, A. Dey, and B. Deb, Phys. Rev. A 110, 062618 (2024)], we identify the total blockade interaction that enters the conditional gate dynamics and state the perturbative conditions under which the second-order Magnus description applies. We then use Floquet-modulated ground-Rydberg drives to construct cnot and controlled-$T$ gates for a prescribed strong-blockade ratio. Numerical simulations show fidelities above 0.998 in the coherent model and reduced sensitivity to the sampled Rabi-frequency and detuning errors. Atomic spontaneous emission lowers the fidelity, with the shorter controlled-$T$ sequence being less affected than the cnot sequence. We distinguish these effective-blockade simulations from a complete experimental error budget: ion heating, excess micromotion, trap-frequency noise, and residual atom-motion entanglement are not included numerically and must be controlled in an implementation. The results establish a concrete connection between ion-mediated blockade enhancement and Floquet-robust conditional control while defining the regime and limitations of the effective model.

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• Geometric & topological phases
• Quantum computation
• Quantum gates
• Qubits
• Rydberg atoms & molecules
• Trapped ions