It is difficult to converge the diffusion coefficient using ab initio molecular dynamics due to its high computational cost, especially when combined with high-level electronic structure theory beyond standard GGA. Diffusion coefficients from high-level electronic structure theory have been only made available by using a multiple time stepping (MTS) method. For instance, Marsalek and Markland carried out revPBE0-D3 calculations using ring-polymer contraction with MTS methods. The calculated diffusion coefficient of liquid water at 300 K is determined to be 2.67 and 2.29 × 10 -5 cm 2 /s from classical and quantum simulations respectively. The inclusion of NQEs decreases the diffusion coefficient by 30%. Del Ben and co-workers combined a hybrid functional (PBEW1-D3) with MP2, using fast (0.25 fs) and slow (2.5 fs) time steps to calculate the dynamical property of MP2 water. Their analysis is based on two trajectories of each roughly 10 ps as obtained in the NVE ensemble, and the self-diffusion constant values obtained are 0.67 and 0.77 × 10 -5 cm 2 /s at 300 and 307 K, respectively. At room temperature, our simulations predict a diffusion constant of 0.629 (0.693) and 0.996 (1.060) × 10 -5 cm 2 /s from classical and quantum dynamics (the quantities in brackets are after finite-size correction ), which is in fair agreement with previous simulations . In fact, a classical MD simulation with a limited timescale of 10-20 ps may yield statistical error bars of more than 20% on dynamical quantities such as the diffusion coefficient. That may explain the minor inconsistency between our values and previous results. Nevertheless, these values are below the experimental diffusion at room temperature, and more similar to a lower-temperature diffusion. In Fig. , we plot the temperature-dependent diffusion coefficient as obtained from classical and quantum dynamics. The diffusion constants are systematically underestimated by MP2 theory for both classical and quantum simulations. When the experimental diffusion constant is shifted to a higher temperature by 40 K, the diffusion constant as calculated from our simulations matches with experimental values. Previous studies show that NQEs contribute a 30% decrease to the self-diffusion of liquid water using revPBE0-D3 functional and a similar slowdown has also been observed using revPBE-D3 functional. Unlike the revPBE0-D3 functional, the diffusion of water upon including NQEs are enhanced by 10-70% from 240 to 320 K, while a slow down of diffusion by 3-7% have been observed beyond 330 K.