This paper proposes fast two-qubit quantum gates for neutral atoms, where the gate operation time is much faster than the time scales associated with 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 paper discusses several schemes for implementing fast quantum gates, focusing on the use of Rydberg states and their interactions. The paper highlights the challenges of designing fast two-qubit gates for neutral atoms, including the difficulty of identifying strong and controllable two-body interactions and the risk of entanglement with motional degrees of freedom. To mitigate these issues, the paper suggests using adiabatic processes to avoid entanglement with motional states, requiring extremely tight traps and low temperatures. The paper presents two models for implementing fast phase gates. Model A uses a sequence of π-pulses to transfer atoms between ground and Rydberg states, with the phase accumulated during the wait time between pulses. Model B uses individual addressing of atoms and a sequence of π and 2π pulses to achieve a phase gate with high fidelity. The adiabatic version of Model B is discussed, which avoids the need for individual addressing and is less sensitive to atomic distance. The paper also discusses decoherence mechanisms, including spontaneous emission, black body radiation, and motional excitation. It calculates the effects of these mechanisms on the fidelity of the gate, showing that they are negligible for the parameters considered. The paper concludes that the proposed schemes allow for robust quantum gates with atoms in lattices that are not filled regularly. The work is supported by various institutions and acknowledges contributions from multiple researchers.This paper proposes fast two-qubit quantum gates for neutral atoms, where the gate operation time is much faster than the time scales associated with 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 paper discusses several schemes for implementing fast quantum gates, focusing on the use of Rydberg states and their interactions. The paper highlights the challenges of designing fast two-qubit gates for neutral atoms, including the difficulty of identifying strong and controllable two-body interactions and the risk of entanglement with motional degrees of freedom. To mitigate these issues, the paper suggests using adiabatic processes to avoid entanglement with motional states, requiring extremely tight traps and low temperatures. The paper presents two models for implementing fast phase gates. Model A uses a sequence of π-pulses to transfer atoms between ground and Rydberg states, with the phase accumulated during the wait time between pulses. Model B uses individual addressing of atoms and a sequence of π and 2π pulses to achieve a phase gate with high fidelity. The adiabatic version of Model B is discussed, which avoids the need for individual addressing and is less sensitive to atomic distance. The paper also discusses decoherence mechanisms, including spontaneous emission, black body radiation, and motional excitation. It calculates the effects of these mechanisms on the fidelity of the gate, showing that they are negligible for the parameters considered. The paper concludes that the proposed schemes allow for robust quantum gates with atoms in lattices that are not filled regularly. The work is supported by various institutions and acknowledges contributions from multiple researchers.