Long-range $CC{\Phi}$ gates via radio-frequency-induced F\"orster resonances

Registers of trapped neutral atoms, excited to Rydberg states to induce strong long-distance interactions, are extensively studied for direct applications in quantum computing. Here, we present a novel $CC\Phi$ quantum phase gate protocol based on radio-frequency-induced F\"{o}rster resonant interactions in the array of highly excited $^{87}$Rb atoms. The extreme controllability of interactions provided by RF field application enables high-fidelity and robust gate performance for a wide range of parameters of the atomic system, as well as it significantly facilitates the experimental implementation of the gate protocol. Taking into account finite Rydberg states lifetimes, we achieve an average theoretical gate fidelity of $99.27 \%$ under room-temperature conditions (improved up to $99.65 \%$ in a cryogenic environment), thus showing the protocol compatibility with modern quantum error correction techniques.