  We present fast algorithms for the summation of Dyson series and the inchworm
Monte Carlo method for quantum systems that are coupled with harmonic baths.
The algorithms are based on evolving the integro-differential equations where
the most expensive part comes from the computation of bath influence
functionals. To accelerate the computation, we design fast algorithms based on
reusing the bath influence functionals computed in the previous time steps to
reduce the number of calculations. It is proven that the proposed fast
algorithms reduce the number of such calculations by a factor of $O(N)$, where
$N$ is the total number of time steps. Numerical experiments are carried out to
show the efficiency of the method and to verify the theoretical results.
