  We study the sign problem in the Hubbard model on the hexagonal lattice away
from half-filling using the Lefschetz thimbles method. We identify the saddle
points, reduce their amount, and perform quantum Monte Carlo (QMC) simulations
using the holomorphic gradient flow to deform the integration contour in
complex space. Finally, the results are compared against exact diagonalization
(ED). We show that the sign problem can be substantially weakened, even in the
regime with low temperature and large chemical potential, where standard QMC
techniques exhibit an exponential decay of the average sign.
