We investigate the Schr$ddot{o}$dinger type operator $H_2;=;(-Delta_{mathbb{H}^n})^2+V^2$ on the Heisenberg group $mathbb{H}^n$, where $Delta_{mathbb{H}^n}$ is the sublaplacian and the nonnegative potential V belongs to the reverse H$ddot{o}$lder class $B_q$ for $qgeqfrac{Q}{2}$, where Q is the homogeneous dimension of $mathbb{H}^n$. We shall establish the estimates of the fundamental solution for the operator $H_2$ and obtain the $L^p$ estimates for the operator $ abla^4_{mathbb{H}^n}H^{-1}_2$, where $ abla_{mathbb{H}^n}$ is the gradient operator on $mathbb{H}^n$.