  Supermassive black holes (SMBHs) of $\sim 10^9\, M_\odot$ are generally
believed to be the central engines of the luminous quasars observed at
$z\gtrsim6$, but their astrophysical origin remains elusive. The $z\gtrsim$
quasars reside in rare density peaks, which poses several challenges to uniform
hydrodynamic simulations. To investigate the formation of these distant
quasars, we perform a suite of zoom-in simulations on a favorable halo, with a
mass of $\sim 10^{13}\, M_\odot$ at $z = 6$ and a history of multiple major
mergers, ideal for BH growth. We test BH seeds of $10 - 10^6\, M_\odot$, and
various accretion and feedback models, including thin-disk and slim-disk
accretion. We find, contrary to previous studies, that light seeds of $\lesssim
10^3\, M_\odot$ fail to grow to $10^8\, M_\odot$ by $z\sim 6$ even with
super-critical accretion; that the hyper-Eddington mode leads to lower
accretion rates than the Eddington-limited case due to stronger feedback,
resulting in significantly smaller BHs by two orders of magnitude; and that
while the super-critical model boosts the growth of low-spin BHs, for high-spin
BHs the mass may be reduced due to increased radiative feedback. Our
simulations show that the first $10^8 - 10^9\, M_\odot$ SMBHs may grow from
heavy seeds of $\gtrsim 10^4\, M_\odot$ via Eddington-limited or mild
super-critical accretion facilitated by gas-rich mergers and self-regulated by
feedback, and they co-evolve with their host galaxies, producing bright quasars
such as those at $z\sim$6 and ULAS J1342+0928, currently the most distant
quasar at z = 7.54.
