  We study the asymptotic behavior of the spectral gap of simple barrier
tunneling problems, which are related to using quantum annealing to find the
global optimum of cost functions defined over n bits. Specifically we look at
the problem of having an n qubit system tunnel through a barrier of width and
height proportional to n^a. We show that with these quantum annealing problems,
the asymptotic, $n\to\infty$, behavior of the spectral gap does not accurately
describe 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 a
feature of simple one-dimensional tunneling problems themselves, casting doubt
on whether asymptotic analysis is an appropriate tool for studying tunneling
problems in quantum annealing for reasonably sized systems.
