Optimal Lower Bound for Ground-State Energy Estimation with a Guiding State

Suppose we can apply the unitary $U=e^{i H}$ for some Hamiltonian $H$, and are given access to a unitary that prepares a guiding state promised to have overlap at least $γ>0$ with the ground space of $H$. Our goal is to estimate the ground-state energy of $H$ within additive error $δ> 0$ and success