Optimal Ground-State Preparation with a Guiding State

Suppose a Hamiltonian $H$ has a unique ground state $\mathrm{\mid }{\psi }_0⟩$ with an eigenvalue $E_0$, and we have an estimate ${\tilde{E}}_0$ such that $\mathrm{\mid }{\tilde{E}}_0-E_0\mathrm{\mid }\le \delta$, and there is a gap of at least $3\delta$ between $E_0$ and all other eigenvalues. Suppose we have a unitary $A$ available that can produce a “guiding state’” $A\mathrm{\mid }0⟩$ that has overlap at least $\gamma$ with $\mathrm{\mid }{\psi }_0⟩$. We show how to obtain an $\epsilon$-approximation of $\mathrm{\mid }{\psi }_0⟩$ using $O(\log (1\mathrm{/}\epsilon )\mathrm{/}\gamma \delta )$ applications of $U=e^{iH}$ and $A$, and their inverses. We give two different algorithms, one based on interleaving amplitude amplification and error-reduction in the style of [HMdW03], and one using the composition of transducers. This paper is the state-preparation follow-up to our two recent ground-state-energy estimation papers [JW26, SdW26]. Combined, our results show an optimal $O(\log (1\mathrm{/}\epsilon )\mathrm{/}\gamma \delta )$ upper bound for ground state preparation.