  We propose a scheme of fast three-qubit Toffoli quantum gate for ultracold
neutral-atom qubits. The scheme is based on the Stark-tuned three-body
F\"{o}rster resonances, which we have observed in our recent experiment
[D.B.Tretyakov et al., Phys.Rev.Lett. 119, 173402 (2017)]. The three-body
resonance corresponds to a transition when the three interacting atoms change
their states simultaneously, and it occurs at a different dc electric field
with respect to the two-body F\"{o}rster resonance. A combined effect of
three-body and two-body F\"{o}rster interactions in external electric and
magnetic fields near the three-body resonance results in complex coherent
behavior of the populations and phases of collective states of a three-atom
system. We have found that it is possible to obtain experimental conditions
suitable to implement three-qubit Toffoli gate with 96.8\% fidelity and less
than 3~$\mu$s duration.
