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Perturbed Hamming weight problems serve as examples of optimization instancesfor which the adiabatic algorithm provably out performs classical simulatedannealing. In this work we study the efficiency of the adiabatic algorithm forsolving the "the Hamming weight with a spike" problem by using several methodsto compute the scaling of the spectral gap at the critical point, which applyfor various ranges of the height and width of the barrier. Our main result is arigorous polynomial lower bound on the minimum spectral gap for the adiabaticevolution when the bit-symmetric cost function has a thin but polynomially highbarrier. This is accomplished by the use of a variational argument with animproved ansatz for the ground state, along with a comparison to the spectrumof the system when no spike term is present. We also give a more detailedtreatment of the spin coherent path-integral instanton method which was used byFarhi, Goldstone, and Gutmann in arXiv:quant-ph/0201031, and consider itsapplicability for estimating the gap for different scalings of barrier heightand width. We adapt the discrete WKB method for an abruptly changing potential,and apply it to the construction of approximate wave functions which can beused to estimate the gap. Finally, the improved ansatz for the ground stateleads to a method for predicting the location of avoided crossings in theexcited states of the energy spectrum of the thin spike Hamiltonian, and we usea recursion relation to determine the ordering of some of these avoidedcrossings, which may be a useful step towards understanding the diabaticcascade phenomenon which occurs in spike Hamiltonians.

The performance of the quantum adiabatic algorithm on spike Hamiltonians