  Eliasson and Kuksin developed a KAM approach to study the persistence of the
invariant tori for nonlinear Schr\"{o}dinger equation on $\mathbb{T}^{d}$. In
this note, we improve Eliasson and Kuksin's KAM theorem by using Kolmogorov's
iterative scheme and obtain a local normal form for the transformed
Hamiltonian. As a consequence, we are able to derive the time $\delta^{-1}$
stability of the obtained KAM tori.
