Stochastic Formalism for Fast Spin-Resolved GW.

We extend the stochastic GW (sGW) formalism to fully spin-polarized systems, encompassing both collinear and noncollinear spin configurations. For noncollinear systems, where Kohn-Sham states are complex two-component spinors, we develop a complex-valued stochastic basis that preserves the real-valued external stochastic charge applied at time zero. This basis enables an unbiased evaluation of the random-phase approximation (RPA) screened interaction for spinors. Through error analysis and tests on real materials, we show that the performance of collinear sGW retains the same time complexity as the spin-unpolarized sGW. The noncollinear sGW incurs a computational cost two to three times higher than the spin-unpolarized version, while preserving linear scaling with low multiplicity. By unifying collinear and noncollinear treatments within a single scalable framework, our work paves the way for routine many-body predictions in large scale magnetic and spin-orbit-coupled material systems.