Theory of Trotter Error with Commutator Scaling
Theory of Trotter Error with Commutator Scaling
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DOI：
10.1103/physrevx.11.011020
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发表时间：
2021-02-01
影响因子：
12.5
通讯作者：
Zhu, Shuchen
中科院分区：
文献类型：
--
作者：
Childs, Andrew M.;Su, Yuan;Zhu, Shuchen
Lie-Trotter公式及其高阶推广提供了一种直接分解算子和的指数的方法。尽管付出了巨大努力，但人们对此类产品配方的误差比例仍然知之甚少。在截断Baker-Campbell-Hausdorff展开式的基础上，我们发展了一种Trotter误差理论，克服了以往方法的局限性。我们的分析直接利用了算符求和的交换性，为实时和虚时演化产生了更紧密的误差界。虽然以前的工作对具有几何局部性或李代数结构的系统实现了类似的目标，但我们的方法总体上是成立的。我们给出了一系列用于数字量子模拟和量子蒙特卡罗方法的改进算法，包括二次量子化的平面波电子结构的模拟、k-局域哈密顿量、快速衰减的幂定律相互作用、簇状哈密顿量、横场伊辛模型和量子铁磁体的模拟，几乎与前人的最好结果相当，甚至超过了最好的结果。利用乘积公式可以保持模拟系统的局部性这一事实，我们获得了进一步的加速比。具体地说，我们证明了对于幂律相互作用系统，局域可观测量可以以与系统大小无关的复杂性被模拟，这意味着作为副产品的Lieb-Robinson界。我们的分析重现了已知的一阶和二阶公式的严格界限。我们的高阶界将模拟一维海森堡模型的复杂性高估了5倍，而对于幂函数相互作用和其他项的排序，它几乎是紧凑的。这一结果表明，我们的理论能够准确地用渐近标度和恒定前因数来刻画Trotter误差。
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.