Super-massive black holes weighing up to $sim 10^9 , mathrm{M_{odot}}$ are in place by $z sim 7$, when the age of the Universe is $lesssim 1 , mathrm{Gyr}$. This implies a time crunch for their growth, since such high masses cannot be easily reached in standard accretion scenarios. Here, we explore the physical conditions that would lead to optimal growth wherein stable super-Eddington accretion would be permitted. Our analysis suggests that the preponderance of optimal conditions depends on two key parameters: the black hole mass and the host galaxy central gas density. In the high-efficiency region of this parameter space, a continuous stream of gas can accrete onto the black hole from large to small spatial scales, assuming a global isothermal profile for the host galaxy. Using analytical initial mass functions for black hole seeds, we find an enhanced probability of high-efficiency growth for seeds with initial masses $gtrsim 10^4 , mathrm{M_{odot}}$. Our picture suggests that a large population of high-$z$ lower-mass black holes that formed in the low-efficiency region, with low duty cycles and accretion rates, might remain undetectable as quasars, since we predict their bolometric luminosities to be $lesssim 10^{41} , mathrm{erg , s^{-1}}$. The presence of these sources might be revealed only via gravitational wave detections of their mergers.
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