Unfortunately this free projection method suffers from numerical instabilities that yield an exponentially growing signal to noise ratio, necessitating the introduction of approximations to keep the simulations tractable. Two strategies are typically used to reduce the variance in the walker weights 27 without introducing a bias: i) the mean-field subtraction L → L-⟨ψ T | L|ψ T ⟩ before the Hubbard-Stratonovich transformation, and ii) the introduction of a shift of each auxiliary field, known as the force-bias, as an importance-sampling transformation. Other algorithmic necessities to retain stability in the imaginary time trajectory are frequent orthonormalization of the orbitals in the walker determinants, and a population control procedure that periodically replaces walkers with small accumulated weights.