  Recently it was shown that the so-called guided local Hamiltonian problem --
estimating the smallest eigenvalue of a $k$-local Hamiltonian when provided
with a description of a quantum state ('guiding state') that is guaranteed to
have substantial overlap with the true groundstate -- is BQP-complete for $k
\geq 6$ when the required precision is inverse polynomial in the system size
$n$, and remains hard even when the overlap of the guiding state with the
groundstate is close to a constant $\left(\frac12 -
\Omega\left(\frac{1}{\mathop{poly}(n)}\right)\right)$. We improve upon this
result in three ways: by showing that it remains BQP-complete when i) the
Hamiltonian is 2-local, ii) the overlap between the guiding state and target
eigenstate is as large as $1 - \Omega\left(\frac{1}{\mathop{poly}(n)}\right)$,
and iii) when one is interested in estimating energies of excited states,
rather than just the groundstate. Interestingly, iii) is only made possible by
first showing that ii) holds.
