Spectral analysis of product formulas for quantum simulation

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

Automated summary

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