= Adiabatic preparation of a computational history state
{title2=$\tau=O(T^6/\epsilon)$}
A Hamiltonian path preserving the history subspace can be analysed using its restricted <spectral gap>. If this gap is $\Omega(T^{-2})$ and the derivative has <operator norm> $O(1)$, a runtime criterion proportional to $\|\dot H\|^2/(\epsilon\Delta^3)$ permits <computational history state> preparation in time $O(T^6/\epsilon)$. Other invariant sectors do not cause leakage from a state initialized in the history subspace.
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