论文标题
Inch Norm Monte Carlo方法的数值分析：标志问题和错误生长
Numerical analysis for inchworm Monte Carlo method: Sign problem and error growth
论文作者
论文摘要
我们考虑了Inchnorm Monte Carlo方法的数值分析，该方法最近提议解决开放量子系统的数值标志问题。我们关注数值误差相对于模拟时间的生长，而对蒙特卡洛方法的直接应用在古典dyson系列中比直接应用的Inch虫蒙特卡洛方法显示出平坦的曲线。为了更好地了解Inch虫蒙特卡洛法的基本机制，我们区分了两种类型的指数误差生长，这些误差生长被称为数值符号问题和误差放大。前者是由于随机方法中方差的快速生长所致，可以从戴森系列中观察到，后者来自数值溶液的演变。我们的分析表明，部分重新召集的技术可以视为平衡这两种错误的工具，而Inchnormmonte Carlo方法是一种成功的情况，在这种情况下，数值符号问题被这种手段有效地抑制了。我们首先在普通微分方程的背景下展示了我们的想法，然后为Inchnorm Monte Carlo方法提供完整的分析。进行了几个数值实验以验证我们的理论结果。
We consider the numerical analysis of the inchworm Monte Carlo method, which is proposed recently to tackle the numerical sign problem for open quantum systems. We focus on the growth of the numerical error with respect to the simulation time, for which the inchworm Monte Carlo method shows a flatter curve than the direct application of Monte Carlo method to the classical Dyson series. To better understand the underlying mechanism of the inchworm Monte Carlo method, we distinguish two types of exponential error growth, which are known as the numerical sign problem and the error amplification. The former is due to the fast growth of variance in the stochastic method, which can be observed from the Dyson series, and the latter comes from the evolution of the numerical solution. Our analysis demonstrates that the technique of partial resummation can be considered as a tool to balance these two types of error, and the inchwormMonte Carlo method is a successful case where the numerical sign problem is effectively suppressed by such means. We first demonstrate our idea in the context of ordinary differential equations, and then provide complete analysis for the inchworm Monte Carlo method. Several numerical experiments are carried out to verify our theoretical results.
