Optimal Ground-State Preparation with a Guiding State

Suppose a Hamiltonian H has a unique ground state ψ 0 with an eigenvalue E 0, and we have an estimate 0 such that 0-E 0 at mostδ, and there is a gap of at least 3δ between E 0 and all other eigenvalues. Suppose we have a unitary A available that can produce a "guiding state'" A0 that has overlap at least γ with ψ 0. We show how to obtain an η-approximation of ψ 0 with probability at least 1-epsilon using O((1/epsilon)/γδ+ (1/η)/δ) applications of U=e iH its inverse, and O((1/epsilon)/γ) applications of A and its inverse. 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 with the optimal energy-estimation of [JW26], our results show the claimed upper bound for ground-state preparation. The bound is optimal up to a constant factor in terms of the number of applications of U and U if epsilon=η, as follows from [SdW26]. In one of our approaches to ground state preparation, we construct transducers for amplitude amplification and LCU that might be of independent interest.