Spectral analysis of product formulas for quantum simulation

Abstract We consider the time-independent Hamiltonian simulation using first order Lie–Trotter–Suzuki product formula under assumption that initial state is supported on a low-dimension subspace. By comparing spectral decomposition of original and effective Hamiltonian, we obtain better upper bounds for various conditions. Especially, show Trotter step size needed to estimate an energy eigenvalue within precision ϵ quantum phase estimation can be improved in scaling from 1/2 large class systems. Our results also depend gap condition simulated Hamiltonian.