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 ϵ using quantum phase estimation can be improved in scaling from ϵ to ϵ^1/2 for a large class of systems. Our results also depend on the gap condition of the simulated Hamiltonian.

中文翻译：

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量子模拟乘积公式的光谱分析

在假设初始状态在低维子空间上得到支持的假设下，我们考虑使用一阶 Lie-Trotter-Suzuki 乘积公式的与时间无关的哈密顿量模拟。通过比较原始哈密顿量和有效哈密顿量的谱分解，我们获得了各种条件下更好的上界。特别是，我们表明，对于一大类系统，使用量子相位估计在精度ε内估计能量特征值所需的 Trotter 步长可以在从ε缩放到ε ^{1/2时得到改进。}我们的结果还取决于模拟哈密顿量的间隙条件。