  The concept of Lefschetz thimble decomposition is one of the most promising
possible modifications of Quantum Monte Carlo (QMC) algorithms aimed at
alleviating the sign problem which appears in many interesting physical
situations, e.g. in the Hubbard model away from half filling. In this approach
one utilizes the fact that the integral over real variables with an integrand
containing a complex fluctuating phase is equivalent to the sum of integrals
over special manifolds in complex space ("Lefschetz thimbles"), each of them
having a fixed complex phase factor. Thus, the sign problem can be reduced if
the resulting sum contains terms with only a few different phases. We explore
the complexity of the sign problem for the few-site Hubbard model on square
lattice combining a semi-analytical study of saddle points and thimbles in
small lattices with several steps in Euclidean time with results of test QMC
calculations. We check different variants of conventional Hubbard-Stratonovich
transformation based on Gaussian integrals. On the basis of our analysis we
reveal a regime with minimal number of relevant thimbles in the vicinity of
half filling. In this regime we found only two relevant thimbles for the
few-site lattices studied in the paper. There is also indirect evidence of the
existence of this regime in more realistic systems with large number of
Euclidean time slices. In addition, we derive a new non-Gaussian representation
of the interaction term, where the number of relevant Lefschetz is also reduced
in comparison with conventional Gaussian Hubbard-Stratonovich transformation.
