  The paper examines the early growth of supermassive black holes (SMBHs) in
cosmological hydrodynamic simulations with different BH seeding scenarios.
Employing the constrained Gaussian realization, we reconstruct the initial
conditions in the large-volume BlueTides simulation and run them to $z=6$ to
cross-validate that the method reproduces the first quasars and their
environments. Our constrained simulations in a volume of $(15\, h^{-1}{\rm
Mpc})^3$ successfully recover the evolution of large-scale structure and the
stellar and BH masses in the vicinity of a $\sim10^{12}\, M_{\odot}$ halo which
we identified in BlueTides at $z\sim7$ hosting a $\sim10^9\, M_{\odot}$ SMBH.
Among our constrained simulations, only the ones with a low-tidal field and
high-density peak in the initial conditions induce the fastest BH growth
required to explain the $z>6$ quasars. We run two sets of simulations with
different BH seed masses of $5\times10^3$, $5\times10^4$, and $5\times10^5\,
h^{-1}M_{\odot}$, (a) with the same ratio of halo to BH seed mass and (b) with
the same halo threshold mass. At $z=6$, all the SMBHs converge in mass to
$\sim10^9\, M_{\odot}$ except for the one with the smallest seed in (b)
undergoing critical BH growth and reaching $10^8$ -- $10^9\, M_{\odot}$, albeit
with most of the growth in (b) delayed compared to set (a). The finding of
eight BH mergers in the small-seed scenario (four with masses $10^4$ -- $10^6\,
M_{\odot}$ at $z>12$), six in the intermediate-seed scenario, and zero in the
large-seed scenario suggests that the vast BHs in the small-seed scenario merge
frequently during the early phases of the growth of SMBHs. The increased BH
merger rate for the low-mass BH seed and halo threshold scenario provides an
exciting prospect for discriminating BH formation mechanisms with the advent of
multi-messenger astrophysics and next-generation gravitational wave facilities.
