Hamiltonian simulations are key subroutines in adiabatic quantum computation, quantum control, and quantum many-body physics, where quantum dynamics often happen in the low-energy sector. In contrast to time-independent Hamiltonian simulations, a comprehensive understanding of quantum simulation algorithms for time-dependent Hamiltonians under the low-energy assumption remains limited hitherto. In this paper, we investigate how much we can improve upon the standard performance guarantee assuming the initial state is supported on a low-energy subspace. In particular, we compute the Trotter number of digital quantum simulation based on product formulas for time-dependent spin Hamiltonians under the low-energy assumption that the initial state is supported on a small number of low-energy eigenstates, and show improvements over the standard cost for simulating full unitary simulations. Technically, we derive the low-energy simulation error with commutator scaling for product formulas by leveraging adiabatic perturbation theory to analyze the time-variant energy spectrum of the underlying Hamiltonian. We further discuss the applications to simulations of non-equilibrium quantum many-body dynamics and adiabatic state preparation. Finally, we prove a lower bound of query complexity for generic time-dependent Hamiltonian simulations.

翻译：哈密顿量模拟是绝热量子计算、量子控制和量子多体物理中的关键子程序，其中量子动力学过程常发生于低能区。与时间无关哈密顿量模拟相比，目前在低能假设下对含时哈密顿量量子模拟算法的系统性理解仍显不足。本文研究了在假设初始态支撑于低能子空间的前提下，我们能在多大程度上改进标准性能保证。具体而言，我们计算了基于乘积公式的数字量子模拟的Trotter数，该模拟针对含时自旋哈密顿量，并基于初始态支撑于少量低能本征态的低能假设，展示了相对于完整幺正模拟标准成本的改进。在技术上，我们通过利用绝热微扰理论分析底层哈密顿量的时变能谱，推导了具有对易子标度的乘积公式低能模拟误差。我们进一步讨论了该方法在非平衡量子多体动力学模拟和绝热态制备中的应用。最后，我们证明了通用含时哈密顿量模拟查询复杂度的下界。