  This study demonstrates that the boundedness of the \( H^\infty \)-calculus
for the negative discrete Laplace operator is independent of the spatial mesh
size. Using this result, we deduce the discrete stochastic maximal \( L^p
\)-regularity estimate for a spatial semidiscretization. Furthermore, we derive
(nearly) sharp error estimates for the semidiscretization under the general
spatial \( L^q \)-norms.
