  In this paper, we consider the large deviations principles (LDPs) for the
stochastic linear Schr\"odinger equation and its symplectic discretizations.
These numerical discretizations are the spatial semi-discretization based on
spectral Galerkin method, and the further full discretizations with symplectic
schemes in temporal direction. First, by means of the abstract G\"artner--Ellis
theorem, we prove that the observable $B_T=\frac{u(T)}{T}$, $T>0$ of the exact
solution $u$ is exponentially tight and satisfies an LDP on $L^2(0, \pi;
\mathbb C)$. Then, we present the LDPs for both $\{B^M_T\}_{T>0}$ of the
spatial discretization $\{u^M\}_{M\in\mathbb N}$ and $\{B^M_N\}_{N\in \mathbb
N}$ of the full discretization $\{u^M_N\}_{M,N\in\mathbb N}$, where
$B^M_T=\frac{u^M(T)}{T}$ and $B^M_N=\frac{u^M_N}{N\tau}$ are the discrete
approximations of $B_T$. Further, we show that both the semi-discretization
$\{u^M\}_{M\in \mathbb N}$ and the full discretization $\{u^M_N\}_{M,N\in
\mathbb N}$ based on temporal symplectic schemes can weakly asymptotically
preserve the LDP of $\{B_T\}_{T>0}$. These results show the ability of
symplectic discretizations to preserve the LDP of the stochastic linear \xde
equation, and first provide an effective approach to approximating the LDP rate
function in infinite dimensional space based on the numerical discretizations.
