  Ultracold-atom simulations of the Hubbard model provide insights into the
character of charge and spin correlations in and out of equilibrium. The
corresponding numerical simulations, on the other hand, remain a significant
challenge. We build on recent progress in the quantum Monte Carlo (QMC)
simulation of electrons in continuous space, and apply similar ideas to the
square-lattice Hubbard model. We devise and benchmark two discrete-time QMC
methods, namely the fermionic-propagator QMC (FPQMC) and the alternating-basis
QMC (ABQMC). In FPQMC, the time evolution is represented by snapshots in real
space, whereas the snapshots in ABQMC alternate between real and reciprocal
space. The methods may be applied to study equilibrium properties within
grand-canonical or canonical ensemble, external field quenches, and even the
evolution of pure states. Various real-space/reciprocal-space correlation
functions are also within their reach. Both methods deal with matrices of size
equal to the number of particles (thus independent of the number of orbitals or
time slices), which allows for cheap updates. We benchmark the methods in
relevant setups. In equilibrium, the FPQMC method is found to have excellent
average sign and, in some cases, yields correct results even with poor
imaginary-time discretization. ABQMC has significantly worse average sign, but
also produces good results. Out of equilibrium, FPQMC suffers from a strong
dynamical sign problem. On the contrary, in ABQMC, the sign problem is not time
dependent. Using ABQMC, we compute survival probabilities for several
experimentally relevant pure states.
