  The discovery of superconductivity in infinite-layer nickelates, with
transition temperature ($T_c$) up to 23 K, provides an exciting new avenue to
study correlated electrons and emergent phases. Superconductivity in the
nickelates has been mostly perceived to be unconventional and originated from
the Ni $d$-electrons due to its analog to cuprate superconductors. The
conventional mechanism for superconductivity - phonon-mediated pairing - was
presumably ruled out because density functional theory (DFT) calculations
reported a very weak electron-phonon coupling in the nickelates. Here, by
including electron self-energy effects on the electronic structure and
electron-phonon coupling (with $\textit{ab initio}$ $GW$ calculations), we
discover that infinite-layer Nd$_{0.8}$Sr$_{0.2}$NiO$_2$ is a dominantly
two-gap phonon-mediated superconductor. We show electron correlations alter the
character of its multi-band Fermi surface and also strongly enhance the
electron-phonon coupling, leading to a large $T_c$ in agreement with
experiment. The computed electron-phonon coupling constant $\lambda$ is
enhanced by an unprecedented factor of 5.5 as compared to DFT. Solutions of the
anisotropic Eliashberg equations yield two dominant $s$-wave gaps - a large gap
on states of rare-earth Nd $d$-electron and interstitial orbital characters but
a small gap on those of transition-metal Ni $d$-electron character. The
superconducting quasiparticle density of states prominently reflects the
two-gap nature and explains well tunneling experiments. Our results demonstrate
that the phonon mechanism accounts for superconductivity in the nickelates,
revealing an unforeseen two-gap $s$-wave nature as well as providing new
insights to demystify the similarities and distinctions between the nickelate
and cuprate superconductors.
