 Simulation of a computable diagonal Hamiltonian

Let $A|x⟩=a(x)|x⟩$, where a reversible circuit $C$ computes $a(x)$ into a clean register. Then applying $e^{-ita(x)}$ as a phase on that register and performing uncomputation implements $e^{-itA}$ on the data register. For $a(x)=(-1{)}^{f(x)}$, a Boolean output qubit suffices: $C^{†}(I\otimes e^{-itZ})C$ acts as $e^{-itA}$ when that qubit starts in $|0⟩$. This exact construction avoids a Lie-Trotter product formula because the eigenvalue is computed directly.