Swift GW beyond 10,000 electrons using sparse stochastic compression

We introduce the concept of sparse stochastic compression, an efficient stochastic sampling of any general function. The technique uses sparse stochastic orbitals (SSOs), short vectors that sample a small number of space points. As a first demonstration, SSOs 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 ${N}_{e}g10,000$ electrons, with small statistical errors of $\ifmmode\pm\else\textpm\fi{}0.05\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$ and using less than 2000 core CPU hours. Overall, stochastic $GW$ scales now linearly (and often sublinearly) with ${N}_{e}.$