Adiabatic preparation of a computational history state ($\tau =O(T^6/ϵ)$)

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 $\Vert \dot{H}{\Vert }^2/(ϵ{\Delta }^3)$ permits computational history state preparation in time $O(T^6/ϵ)$. Other invariant sectors do not cause leakage from a state initialized in the history subspace.