  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.
