  In this review we consider the performance of the quantum adiabatic algorithm
for the solution of decision problems. We divide the possible failure
mechanisms into two sets: small gaps due to quantum phase transitions and small
gaps due to avoided crossings inside a phase. We argue that the thermodynamic
order of the phase transitions is not predictive of the scaling of the gap with
the system size. On the contrary, we also argue that, if the phase surrounding
the problem Hamiltonian is a Many-Body Localized (MBL) phase, the gaps are
going to be typically exponentially small and that this follows naturally from
the existence of local integrals of motion in the MBL phase.
