  The Trotter decomposition is a basic approach to Hamiltonian simulation
(digital quantum simulation). The first-order Trotter decomposition is the
simplest one, whose deviations from target dynamics are of the first order of a
small coefficient in terms of the infidelity. In this paper, we consider the
first-order Trotter decomposition in the dynamical-invariant basis. By using a
state-dependent inequality, we point out that deviations of this decomposition
are of the second order of a small coefficient. Moreover, we also show that
this decomposition includes a useful example, i.e., digital implementation of
shortcuts to adiabaticity by counterdiabatic driving.
