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Success of digital adiabatic simulation with large Trotter step

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.