  The algorithm based on integration over Lefschetz thimbles is a promising
method to resolve the sign problem for complex actions. However, this algorithm
often meets a difficulty in actual Monte Carlo calculations because the
configuration space is not easily explored due to the infinitely high potential
barriers between different thimbles. In this paper, we propose to use the flow
time of the antiholomorphic gradient flow as an auxiliary variable for the
highly multimodal distribution. To illustrate this, we implement the parallel
tempering method by taking the flow time as a tempering parameter. In this
algorithm, we can take the maximum flow time to be sufficiently large such that
the sign problem disappears there, and two separate modes are connected through
configurations at small flow times. To exemplify that this algorithm does work,
we investigate the (0+1)-dimensional massive Thirring model at finite density
and show that our algorithm correctly reproduces the analytic results for large
flow times such as T=2.
