  The rapid assembly of the massive black holes that power the luminous quasars
observed at $z \sim 6-7$ remains a puzzle. Various direct collapse models have
been proposed to head-start black hole growth from initial seeds with masses
$\sim 10^5\,\rm M_\odot$, which can then reach a billion solar mass while
accreting at the Eddington limit. Here we propose an alternative scenario based
on radiatively inefficient super-critical accretion of stellar-mass holes
embedded in the gaseous circum-nuclear discs (CNDs) expected to exist in the
cores of high redshift galaxies. Our sub-pc resolution hydrodynamical
simulations show that stellar-mass holes orbiting within the central 100 pc of
the CND bind to very high density gas clumps that arise from the fragmentation
of the surrounding gas. Owing to the large reservoir of dense cold gas
available, a stellar-mass black hole allowed to grow at super-Eddington rates
according to the "slim disc" solution can increase its mass by 3 orders of
magnitudes within a few million years. These findings are supported by
simulations run with two different hydro codes, RAMSES based on the Adaptive
Mesh Refinement technique and GIZMO based on a new Lagrangian Godunov-type
method, and with similar, but not identical, sub-grid recipes for star
formation, supernova feedback, black hole accretion and feedback. The low
radiative efficiency of super-critical accretion flows are instrumental to the
rapid mass growth of our black holes, as they imply modest radiative heating of
the surrounding nuclear environment.
