= Solution
Write the <local Hamiltonian> as $H=\sum_{j=1}^mh_j$. Since its terms commute, their <matrix exponentials> factor exactly:
$$
e^{-iHt}=\prod_{j=1}^me^{-ih_jt}.
$$
Each factor acts on at most two qubits and can be compiled over a fixed <universal quantum gate set> to <operator norm> error at most $\varepsilon/m$. The <telescoping bound for products of operators> then bounds the total error by the sum of the factor errors, at most $\varepsilon$. Because $m$ is polynomial in $n$ and the <Solovay--Kitaev theorem> gives gate count polynomial in $\log(m/\varepsilon)$ for each fixed-dimensional factor, this is an efficient <commuting local Hamiltonian simulation>. Finally, the <eigenvalue equation> $H|\Psi\rangle=\lambda|\Psi\rangle$ implies
$$
U(t)|\Psi\rangle=e^{-i\lambda t}|\Psi\rangle,
$$
so $|\Psi\rangle$ remains an <eigenstate> and its eigenvalue is $e^{-i\lambda t}$.
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