Continuous-time quantum search ($H=\Delta (P_s+P_w),t_{\ast }=\pi /(2\Delta |⟨s|w⟩|)$)

Take normalized states with real positive overlap $a\in (0,1)$. The eigenvectors proportional to $|s⟩\pm |w⟩$ have energies $\Delta (1\pm a)$. With $\hslash =1$, their relative phase changes by $\pi$ after the displayed time, giving $e^{-iH{t}_{\ast }}|s⟩=-i{e}^{-i\Delta t_{\ast }}|w⟩$. 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.