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The so-called phaseless quantum Monte-Carlo method currently offers one ofthe best performing theoretical framework to investigate interacting Fermisystems. It allows to extract an approximate ground-state wavefunction byaveraging independent-particle states undergoing a Brownian motion inimaginary-time. Here, we extend the approach to a random walk in the space ofHartree-Fock-Bogoliubov (HFB) vacua that are better suited for superconductingor superfluid systems. Well-controlled statistical errors are ensured byconstraining stochastic paths with the help of a trial wavefunction, alsoguiding the dynamics and in the form of a linear combination of HFB ans\"atze.Estimates for the observables are reconstructed through an extension of Wick'stheorem to matrix elements between HFB product states. The usual combinatorycomplexity associated to the application of this theorem for four- and more-body operators is bypassed with a compact expression in terms of Pfaffians. Thelimiting case of a stochastic motion within Slater determinants but guided withHFB trial wavefunctions is also considered. Finally, exploratory results forthe spin polarized Hubbard model in the attractive regime are presented.

Phaseless quantum Monte-Carlo approach to strongly correlated superconductors with stochastic Hartree-Fock-Bogoliubov wavefunctions