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The effect of local minima on quantum adiabatic optimization
We present a perturbative method to estimate the spectral gap for quantum adiabatic 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 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.