  We consider symmetry-projected Hartree--Fock trial wave functions in
constrained-path Monte Carlo (CPMC) calculations. Previous CPMC calculations
have mostly employed Hartree--Fock (HF) trial wave functions, restricted or
unrestricted. The symmetry-projected HF approach results in a hierarchy of wave
functions with increasing quality: the more symmetries that are broken and
restored in a self-consistent manner, the higher the quality of the trial wave
function. This hierarchy is approximately maintained in CPMC calculations: the
accuracy in the energy increases and the statistical variance decreases when
further symmetries are broken and restored. Significant improvement is achieved
in CPMC with the best symmetry-projected trial wave functions over those from
simple HF. We analyze and quantify the behavior using the two-dimensional
repulsive Hubbard model as an example. In the sign-problem-free region, where
CPMC can be made exact but a constraint is deliberately imposed here,
spin-projected wave functions remove the constraint bias. Away from
half-filling, spatial symmetry restoration in addition to that of the spin lead
to highly accurate results from CPMC. Since the computational cost of
symmetry-projected HF trial wave functions in CPMC can be made to scale
algebraically with system size, this provides a potentially general approach
for accurate calculations in many-fermion systems.
