  We prove the reducibility of quantum harmonic oscillators in $\mathbb R^d$
perturbed by a quasi-periodic in time potential $V(x,\omega t)$ with
$\mathit{logarithmic~decay}$. By a new estimate built for solving the
homological equation we improve the reducibility result by
Gr\'ebert-Paturel(Annales de la Facult\'e des sciences de Toulouse :
Math\'ematiques. $\mathbf{28}$, 2019).
