  We present a perturbative method to estimate the spectral gap for adiabatic
quantum optimization, based on the structure of the energy levels in the
problem Hamiltonian. We show that for problems that have exponentially large
number of local minima close to the global minimum, the gap becomes
exponentially small making the computation time exponentially long. The quantum
advantage of adiabatic quantum computation may then be accessed only via the
local adiabatic evolution, which requires phase coherence throughout the
evolution and knowledge of the spectrum. Such problems, therefore, are not
suitable for adiabatic quantum computation.
