  The simulation of adiabatic evolution has deep connections with Adiabatic
Quantum Computation, the Quantum Approximate Optimization Algorithm and
adiabatic state preparation. Here we address the error analysis problem in
quantum simulation of adiabatic process using Trotter formulas. We show that
with additional conditions, the circuit depth can be linear in simulation time
$T$. The improvement comes from the observation that the fidelity error here
can't be estimated by the norm distance between evolution operators. This
phenomenon is termed as the robustness of discretization in digital adiabatic
simulation. It can be explained in three steps, from analytical and numerical
evidence: (1). The fidelity error should be estimated by applying adiabatic
theorem on the effective Hamiltonian instead. (2). Because of the specialty of
Riemann-Lebesgue lemma, most adiabatic process is naturally robust against
discretization. (3). As the Trotter step gets larger, the spectral gap of
effective Hamiltonian tends to close, which results in the failure of digital
adiabatic simulation.
