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We introduce the concept of fractured stochastic orbitals (FSOs), shortvectors that sample a small number of space points and enable an efficientstochastic sampling of any general function. As a first demonstration, FSOs areapplied in conjunction with simple direct-projection to accelerate our recentstochastic $GW$ technique; the new developments enable accurate prediction of$G_{0}W_{0}$ quasiparticle energies and gaps for systems with up to$N_{e}>10,000$ electrons, with small statistical errors of $\pm0.05\,{\rm eV}$and using less than 2000 core CPU hours. Overall, stochastic $GW$ scales nowlinearly (and often sub-linearly) with $N_{e}.$

Swift $GW$ beyond $10,000$ electrons using fractured stochastic orbitals