  We establish convergence results related to the operator splitting scheme on
the Cauchy problem for the nonlinear Schr\"odinger equation with rough initial
data in $L^2$,
  $$
  \left\{
  \begin{array}{ll}
  i\partial_t u +\Delta u = \lambda |u|^{p} u, & (x,t) \in \mathbb{R}^d \times
\mathbb{R}_+,
  u (x,0) =\phi (x), & x\in\mathbb{R}^d,
  \end{array}
  \right.
  $$ where $\lambda \in \{-1,1\}$ and $p >0$. While the Lie approximation $Z_L$
is known to converge to the solution $u$ when the initial datum $\phi$ is
sufficiently smooth, the convergence result for rough initial data is open to
question. In this paper, for rough initial data $\phi\in L^2 (\mathbb{R}^d)$,
we prove the convergence of the filtered Lie approximation $Z_{flt}$ to the
solution $u$ in the mass-subcritical range, $\max\left\{1,\frac{2}{d}\right\}
\leq p < \frac{4}{d}$. Furthermore, we provide a precise convergence result for
radial initial data $\phi\in L^2 (\mathbb{R}^d)$,
