Fermionic quantum Monte Carlo (QMC) methods based on sampling in real space. Well-known fermion sign problems in simulations of quantum systems are circumvented by the fixed-node/phase approximations with corresponding biases that are, in general, surprisingly small. The resulting QMC approaches show two key advantages: i) variational formulation with complete (stochastic) basis from the outset; ii) mapping of the fermionic problem onto a bosonic one with state-specific, many-body, effective potential that is systematically improvable. Examples of such calculations encompass electronic structure of strongly correlated systems, molecules, solids, ultracold atomic condensates, etc. So far, however, such QMC calculations have been limited to static, collinear treatment of electronic spins. Recently, we introduced QMC with spin-dependent Hamiltonians such as spin-orbit. Interestingly, the constructed spinor-based wave functions with fixed-phase approximation contain the fixed-node solution as a special case and
therefore cover both cases as particular limits. The method also opens further
prospects to improve upon fixed-phase biases progressing therefore towards
nearly exact results.