The paper proposes several schemes for implementing fast two-qubit quantum gates for neutral atoms, with the gate operation time significantly shorter than the external motion of the atoms in the trapping potential. The large interaction energy required for fast gate operations is provided by the dipole-dipole interaction of atoms excited to low-lying Rydberg states in constant electric fields. The authors analyze the imperfections of the gate operation, including decoherence mechanisms such as spontaneous emission, transitions induced by blackbody radiation, ionization of Rydberg states, and motional excitation of trapped atoms. They present two models for phase gates: Model A, which uses a strong Rabi frequency to achieve a phase gate with a time scale proportional to the inverse of the Rabi frequency, and Model B, an adiabatic version that avoids population in the doubly excited state and is less sensitive to atomic distance. The paper also discusses the robustness of the gates and the probabilities of various decoherence processes.The paper proposes several schemes for implementing fast two-qubit quantum gates for neutral atoms, with the gate operation time significantly shorter than the external motion of the atoms in the trapping potential. The large interaction energy required for fast gate operations is provided by the dipole-dipole interaction of atoms excited to low-lying Rydberg states in constant electric fields. The authors analyze the imperfections of the gate operation, including decoherence mechanisms such as spontaneous emission, transitions induced by blackbody radiation, ionization of Rydberg states, and motional excitation of trapped atoms. They present two models for phase gates: Model A, which uses a strong Rabi frequency to achieve a phase gate with a time scale proportional to the inverse of the Rabi frequency, and Model B, an adiabatic version that avoids population in the doubly excited state and is less sensitive to atomic distance. The paper also discusses the robustness of the gates and the probabilities of various decoherence processes.