The Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of prior approaches based on truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds, in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, including simulations of second-quantized plane-wave electronic structure, $k$-local Hamiltonians, rapidly decaying power-law interactions, clustered Hamiltonians, the transverse field Ising model, and quantum ferromagnets, nearly matching or even outperforming the best previous results. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound as a by-product. Our analysis reproduces known tight bounds for first- and second-order formulas. Our higher-order bound overestimates the complexity of simulating a one-dimensional Heisenberg model with an even-odd ordering of terms by only a factor of 5, and it is close to tight for power-law interactions and other orderings of terms. This result suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor.

中文翻译：

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换向器定标的托特误差理论

Lie-Trotter公式及其高阶泛化为分解算子和的指数提供了直接的方法。尽管付出了巨大的努力，但对此类产品公式的误差定标仍然知之甚少。我们开发了一种Trotter错误理论，该理论克服了基于截断Baker-Campbell-Hausdorff展开式的现有方法的局限性。我们的分析直接利用了算子求和式的可交换性，从而为实时和虚时演化产生了更严格的误差范围。尽管先前的工作对于具有几何局部性或李-代数结构的系统也达到了类似的目标，但通常我们的方法仍然适用。我们为数字量子仿真和量子蒙特卡洛方法提供了许多改进的算法，$ķ$-局部哈密顿量，迅速衰减的幂律相互作用，聚集的哈密顿量，横向场伊辛模型和量子铁磁体，几乎与甚至超过了以前的最佳结果。我们使用乘积公式可以保留模拟系统的局部性这一事实，进一步提高了速度。具体来说，我们表明可以用复杂度来模拟局部可观测值，而与幂律交互系统的系统大小无关，这意味着副产品必然是Lieb-Robinson绑定。我们的分析再现了一阶和二阶公式的已知紧界。我们的高阶边界高估了用奇数项的奇数顺序模拟一维Heisenberg模型的复杂度，仅为5倍，对于幂律相互作用和其他项的顺序来说，它接近严格。