  We consider the derivation of the defocusing cubic nonlinear Schr\"{o}dinger
equation (NLS) on $\mathbb{R}^{3}$ from quantum $N$-body dynamics. We reformat
the hierarchy approach with Klainerman-Machedon theory and prove a
bi-scattering theorem for the NLS to obtain convergence rate estimates under
$H^{1}$ regularity. The $H^{1}$ convergence rate estimate we obtain is almost
optimal for $H^{1}$ datum, and immediately improves if we have any extra
regularity on the limiting initial one-particle state.
