We study the problem of simulating the dynamics of spin systems when the initial state is supported on a subspace of low energy of a Hamiltonian H. This is a central problem in physics with vast applications in many-body systems and beyond, where the interesting physics takes place in the low-energy sector. We analyze error bounds induced by product formulas that approximate the evolution operator and show that these bounds depend on an effective low-energy norm of H. We find improvements over the best previous complexities of product formulas that apply to the general case, and these improvements are more significant for long evolution times that scale with the system size and/or small approximation errors. To obtain these improvements, we prove exponentially decaying upper bounds on the leakage to high-energy subspaces due to the product formula. Our results provide a path to a systematic study of Hamiltonian simulation at low energies, which will be required to push quantum simulation closer to reality.

我们研究了当初始状态由哈密顿量H 的低能量子空间支持时模拟自旋系统动力学的问题。这是物理学中的一个核心问题，在多体系统及其他系统中有着广泛的应用，其中有趣的物理学发生在低能量领域。我们分析了由近似演化算子的​​乘积公式引起的误差界限，并表明这些界限取决于H的有效低能量范数. 我们发现对适用于一般情况的产品公式的最佳先前复杂性的改进，并且这些改进对于随着系统大小和/或小的近似误差而缩放的长演化时间更为显着。为了获得这些改进，我们证明由于乘积公式，泄漏到高能子空间的上限呈指数衰减。我们的结果为系统研究低能下的哈密顿模拟提供了一条途径，这将需要推动量子模拟更接近现实。