We study the asymptotic behavior of the spectral gap of simple barriertunneling problems, which are related to using quantum annealing to find theglobal optimum of cost functions defined over n bits. Specifically we look atthe problem of having an n qubit system tunnel through a barrier of width andheight proportional to n^a. We show that with these quantum annealing problems,the asymptotic, $n\to\infty$, behavior of the spectral gap does not accuratelydescribe the behavior of the gap at finite n until extremely large values of n(n>10^{12}). We prove that this deficiency of the asymptotic expression is afeature of simple one-dimensional tunneling problems themselves, casting doubton whether asymptotic analysis is an appropriate tool for studying tunnelingproblems in quantum annealing for reasonably sized systems.
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