  The generalized Lefschetz thimble method is a promising approach that
attempts to solve the sign problem in Monte Carlo methods by deforming the
integration contour using the flow equation. Here we point out a general
problem that occurs due to the property of the flow equation, which extends a
region on the original contour exponentially to a region on the deformed
contour. Since the growth rate for each eigenmode is governed by the singular
values of the Hessian of the action, a huge hierarchy in the singular value
spectrum, which typically appears for large systems, leads to various technical
problems in numerical simulations. We solve this hierarchical growth problem by
preconditioning the flow so that the growth rate becomes identical for every
eigenmode. As an example, we show that the preconditioned flow enables us to
investigate the real-time quantum evolution of an anharmonic oscillator with
the system size that can hardly be achieved by using the original flow.
