  The path integral formulation of quantum mechanical problems including
fermions is often affected by a severe numerical sign problem. We show how such
a sign problem can be alleviated by a judiciously chosen constant imaginary
offset to the path integral. Such integration contour deformations introduce no
additional computational cost to the Hybrid Monte Carlo algorithm, while its
effective sample size is greatly increased. This makes otherwise unviable
simulations efficient for a wide range of parameters. Applying our method to
the Hubbard model, we find that the sign problem is significantly reduced.
Furthermore, we prove that it vanishes completely for large chemical
potentials, a regime where the sign problem is expected to be particularly
severe without imaginary offsets. In addition to a numerical analysis of such
optimized contour shifts, we analytically compute the shifts corresponding to
the leading and next-to-leading order corrections to the action. We find that
such simple approximations, free of significant computational cost, suffice in
many cases.
