  Understanding NP-complete problems is a central topic in computer science.
This is why adiabatic quantum optimization has attracted so much attention, as
it provided a new approach to tackle NP-complete problems using a quantum
computer. The efficiency of this approach is limited by small spectral gaps
between the ground and excited states of the quantum computer's Hamiltonian. We
show that the statistics of the gaps can be analyzed in a novel way, borrowed
from the study of quantum disordered systems in statistical mechanics. It turns
out that due to a phenomenon similar to Anderson localization, exponentially
small gaps appear close to the end of the adiabatic algorithm for large random
instances of NP-complete problems. This implies that unfortunately, adiabatic
quantum optimization fails: the system gets trapped in one of the numerous
local minima.
