标题： Trotterization 对低能态来说是相当高效的 显示英文标题

标题： Trotterization is substantially efficient for low-energy states

摘要： Trotterization是量子计算机或张量网络上模拟量子多体动力学的核心方法之一。 除了其实现简单之外，最近的研究表明，如果初始状态处于低能子空间中，其误差和成本可以降低。 然而，在以前的研究中，通过低能性质带来的改进随着Trotter阶数的增加而迅速消失，因此，是否存在低能初始状态的真实优势仍然是一个谜。 在本信中，我们通过证明低能初始状态的Trotterization的最佳误差界和成本来解决这个问题。 对于由半正定项组成的通用局部哈密顿量，我们表明Trotter误差最多与初始状态能量$\Delta$成线性关系，并且与系统大小$N$呈对数多项式关系。 因此，对于具有$\Delta \in o(Ng)$的低能态，计算成本相比任意初始状态显著减少，其中$g$表示每格点的能量，$Ng$表示整个系统的能量。 我们的Trotterization误差界和成本在初始状态能量$\Delta$上实现了理论上最佳的缩放。 此外，它们可以部分扩展到具有低能期望值的弱关联初始状态，这些状态不一定封闭在低能子空间中。 我们的结果将为快速准确地模拟低能态铺平道路，这些状态是凝聚态物理和量子化学中的一个核心目标。 显示英文摘要

摘要： Trotterization is one of the central approaches for simulating quantum many-body dynamics on quantum computers or tensor networks. In addition to its simple implementation, recent studies have revealed that its error and cost can be reduced if the initial state is closed in the low-energy subspace. However, the improvement by the low-energy property rapidly vanishes as the Trotter order grows in the previous studies, and thus, it is mysterious whether there exists genuine advantage of low-energy initial states. In this Letter, we resolve this problem by proving the optimal error bound and cost of Trotterization for low-energy initial states. For generic local Hamiltonians composed of positive-semidefinite terms, we show that the Trotter error is at most linear in the initial state energy $\Delta$ and polylogarithmic in the system size $N$. As a result, the computational cost becomes substantially small for low-energy states with $\Delta \in o(Ng)$ compared to the one for arbitrary initial states, where $g$ denotes the energy per site and $Ng$ means the whole-system energy. Our error bound and cost of Trotterization achieve the theoretically-best scaling in the initial state energy $\Delta$. In addition, they can be partially extended to weakly-correlated initial states having low-energy expectation values, which are not necessarily closed in the low-energy subspace. Our results will pave the way for fast and accurate simulation of low-energy states, which are one central targets in condensed matter physics and quantum chemistry.