We explain why quantum adiabatic evolution and simulated annealing perform similarly in certain examples of searching for the minimum of a cost function of n bits. In these examples each bit is treated symmetrically so the cost function depends only on the Hamming weight of the n bits. We also give two examples, closely related to these, where the similarity breaks down in that the quantum adiabatic algorithm succeeds in polynomial time whereas simulated annealing requires exponential time.
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Farhi et al. (2002) studied this question.