= Continuous-time quantum search
{title2=$H=\Delta(P_s+P_w),\quad t_* =\pi/(2\Delta|\langle s|w\rangle|)$}
Take normalized states with real positive overlap $a\in(0,1)$. The eigenvectors proportional to $|s\rangle\pm|w\rangle$ have energies $\Delta(1\pm a)$. With $\hbar=1$, their relative phase changes by $\pi$ after the displayed time, giving $e^{-iHt_*}|s\rangle=-ie^{-i\Delta t_*}|w\rangle$. For a uniform search state and one marked basis vector in dimension $D$, $a=D^{-1/2}$, so the time is proportional to $\sqrt D$. Implementing the marked projector assumes access to a search oracle; the formula does not supply oracle information for free.
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