  The Margolus-Levitin (ML) bound says that for any time-independent
Hamiltonian, the time needed to evolve from one quantum state to another is at
least $\pi \alpha(\epsilon) / (2 \langle E-E_0 \rangle)$, where $\langle E-E_0
\rangle$ is the expected energy of the system relative to the ground state of
the 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 known
although 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 the
ML bound, and gave this bound a symplectic-geometric interpretation. Here I
solve the same problem through an elementary proof of the ML bound. By
explicitly finding all the states that saturate the ML bound, I show that
$\alpha_{>}(\epsilon)$ is indeed equal to $\alpha_{<}(\epsilon)$. More
importantly, I point out a numerical stability issue in computing
$\alpha_{>}(\epsilon)$ and report a simple way to evaluate it efficiently and
accurately.

Quantum Physics