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

We introduce the concept of fractured stochastic orbitals (FSOs), short vectors that sample a small number of space points and enable an efficient stochastic sampling of any general function. As a first demonstration, FSOs are applied in conjunction with simple direct-projection to accelerate our recent stochastic $GW$ 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 $\pm 0.05\mathrm{e}\mathrm{V}$ and using less than 2000 core CPU hours. Overall, stochastic $GW$ scales now linearly (and often sub-linearly) with $N_e\mathrm{.}$