  Trotter decomposition is the basis of the digital quantum simulation.
Asymmetric and symmetric decompositions are used in our GPU demonstration of
the digital adiabatic quantum simulations of $2+1$ dimensional quantum
$\mathbb{Z}_2$ lattice gauge theory. The actual errors in Trotter
decompositions are investigated as functions of the coupling parameter and the
number of Trotter substeps in each step of the variation of coupling parameter.
The relative error of energy is shown to be closely related to the Trotter
error usually defined defined in terms of the evolution operators. They are
much smaller than the order-of-magnitude estimation. The error in the symmetric
decomposition is much smaller than that in the asymmetric decomposition. The
features of the Trotter errors obtained here are useful in the experimental
implementation of digital quantum simulation and its numerical demonstration.
