  The tempered Lefschetz thimble method (TLTM) is a parallel-tempering
algorithm towards solving the numerical sign problem. It tames both the sign
and ergodicity problems simultaneously by tempering the system with the flow
time of continuous deformations of the integration region. In this article,
after reviewing the basics of the TLTM, we explain a new algorithm within the
TLTM that enables us to estimate the expectation values precisely with a
criterion ensuring global equilibrium and the sufficiency of the sample size.
To demonstrate the effectiveness of the algorithm, we apply the TLTM to the
quantum Monte Carlo simulation of the Hubbard model away from half filling on a
two-dimensional lattice of small size, and show that the obtained numerical
results agree nicely with exact values.
