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The Margolus-Levitin (ML) bound says that for any time-independentHamiltonian, the time needed to evolve from one quantum state to another is atleast $\pi \alpha(\epsilon) / (2 \langle E-E_0 \rangle)$, where $\langle E-E_0\rangle$ is the expected energy of the system relative to the ground state ofthe Hamiltonian and $\alpha(\epsilon)$ is a function of the fidelity $\epsilon$between the two state. For a long time, only a upper bound$\alpha_{>}(\epsilon)$ and lower bound $\alpha_{<}(\epsilon)$ are knownalthough they agree up to at least seven significant figures. Lately,H\"{o}rnedal and S\"{o}nnerborn proved an analytical expression for$\alpha(\epsilon)$, fully classified systems whose evolution times saturate theML bound, and gave this bound a symplectic-geometric interpretation. Here Isolve the same problem through an elementary proof of the ML bound. Byexplicitly finding all the states that saturate the ML bound, I show that$\alpha_{>}(\epsilon)$ is indeed equal to $\alpha_{<}(\epsilon)$. Moreimportantly, I point out a numerical stability issue in computing$\alpha_{>}(\epsilon)$ and report a simple way to evaluate it efficiently andaccurately.

$\boldsymbol{\alpha_{>}(\epsilon) = \alpha_{<}(\epsilon)}$ For The Margolus-Levitin Quantum Speed Limit Bound