  Neutron stars can be destroyed by black holes at their center accreting
material and eventually swallowing the entire star. Here we note that the
accretion model adopted in the literature, based on Bondi accretion or
variations thereof, is inadequate for small black holes -- black holes whose
Schwarzschild radius is comparable to, or smaller than, the neutron's de
Broglie wavelength. In this case, quantum mechanical aspects of the accretion
process cannot be neglected, and give rise to a completely different accretion
rate. We show that for the case of black holes seeded by the collapse of
bosonic dark matter, this is the case for electroweak-scale dark matter
particles. In the case of fermionic dark matter, typically the black holes that
would form at the center of a neutron star are more massive, unless the dark
matter particle mass is very large, larger than about 10$^{10}$ GeV. We
calculate the lifetime of neutron stars harboring a ``small'' black hole, and
find that black holes lighter than $\sim 10^{11}$ kg quickly evaporate, leaving
no trace. More massive black holes destroy neutron stars via quantum accretion
on time-scales much shorter than the age of observed neutron stars. We find
that the range where seed black holes inside neutron stars are massive enough
that they do not quickly evaporate away, but not so massive that a fluid
accretion picture is warranted is limited to between $\sim10^{11}$ and
$10^{12}$ kg, but our results are key to accurately determine the actual
critical black hole mass corresponding to the onset of neutron star destruction
