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 neutrons 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.
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