  We study the two dimensional Hubbard model by use of the ground state
algorithm in the Monte Carlo simulation. We employ complex wave functions as
trial function in order to have a close look at properties such as chiral spin
order ($\chi$SO) and flux phase. For half filling, a particle-hole
transformation leads to sum rules with respect to the Green's functions for a
certain choice of a set of wave functions. It is then analytically shown that
the sum rules lead to the absence of the $\chi$SO. Upon doping, we are
confronted with the sign problem, which in our case %ch appears as a ``phase
problem" due to the phase of the Monte Carlo weights. The average of the phase
shows an exponential decay as a function of inverse temperature similarly to
that of sign by Loh Jr. et. al. . We compare the numerical results with those
of exact numerical calculations.
