Denominator: 2(langle Hrangle - E_0)

The passage around this formula

The baseline is the quantum speed limit. For a system with a fixed, time-independent Hamiltonian H evolving from a state $\lvert$$\psi$$\rangle$ toward any state orthogonal to it, the minimum time required is bounded below by the tight, unified Margolus-Levitin/Mandelstam-Tamm result of Levitin and Toffoli: $\tau_\perp \geq \frac{\pi\hbar}{2\left(\langle H\rangle - E_0\right)}$. where $\langle$ H$\rangle$ is the mean energy in the evolving state and $E_0$ is the ground-state energy of H [ 2 , 1 ] . Inverting this gives a maximum rate of orthogonal state changes per unit proper time, 2($\langle$ H$\rangle$-$E_0$)/$\pi$$\hbar$ , and Giovannetti, Lloyd, and Maccone showed that for a periodic evolution this rate integrates cleanly: a system with mean energy $\langle$…

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