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

Abstract

The guided Hamiltonian problem is the following: given access to the unitary $U=e^{i H}$ for some Hamiltonian $H$, and given access to a unitary that prepares a guiding state promised to have overlap at least $\gamma >0$ with the ground space of $H$, estimate the ground-state energy of $H$ within additive error $\delta > 0$ and success probability at least $1-\varepsilon $, $\varepsilon>0$. How many applications of $U$ and its inverse $U^{-1}$ are necessary and sufficient? An upper bound $O(\log(1/\varepsilon)\log(1/\gamma)/\gamma\delta)$ was known, and was improved to $O(\log(1/\varepsilon)/\gamma\delta)$ very recently [JW26]. A matching lower bound was known whenever one of the three parameters $\delta,\gamma,\varepsilon$ was held constant [MdW26]. In this paper we prove the joint lower bound $\Omega(\log(1/\varepsilon)/\gamma\delta)$ with the tight $\varepsilon$-dependence provided the dimension of $H$ is at least $\log(1/\varepsilon)/\gamma^2$. Furthermore, we show that this same lower bound (with slightly larger dimension) holds for both the special case in which the ground state is guaranteed to be unique and $H$ has a gap of $\delta$ between its first and second eigenvalue; and for ground-state preparation, where $\delta$ denotes the spectral gap and $\varepsilon$ now is the approximation error. The lower bounds also apply when the Hamiltonian can be accessed via its block-encoding, and when fractional powers of $U$ are allowed, as in continuous-time Hamiltonian simulation. Lastly, improved upper bounds are known when $H$ is nonnegative and presented as a sum of squares; and our results imply the lower bound $\Omega(\log(1/\varepsilon)/\gamma\sqrt{\delta})$ for this case.