Database search can be looked upon as Hamiltonian evolution, with kinetic and potential energy operators. Let the evolution follow the Schrodinger equation:

$$i\frac{d}{dt}|\psi⟩= H|ψ⟩$$ with $H = E|s⟩⟨s| + E|t⟩⟨t|$ and some constant $E$. How can we find the minimum time $T$ required for the initial state $|s⟩ = \frac{1}{\sqrt{N}}\sum_{i=1}^N |i⟩ $ to evolve $N$ to the final state $|t⟩$.

  • $\begingroup$ So do you mean $|s\rangle$ and $|t\rangle$ have the same eigenvalue? $\endgroup$
    – Zhibo Yang
    Oct 23 at 18:24
  • $\begingroup$ @ZhiboYang Yes the eigen value is going to be same $\endgroup$ Oct 24 at 9:53
  • 1
    $\begingroup$ What did you try so far? Did you calculate the unitary time evolution? Apply it to the initial state? $\endgroup$
    – M. Stern
    Oct 24 at 12:10
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    $\begingroup$ It sounds like you are asking how long adiabatic evolution takes. This is given by the spectral gap of the Hamiltonian from start to finish. $\endgroup$
    – Mark S
    Oct 27 at 3:34
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    $\begingroup$ The usual series for the exponential function also works with matrices $\endgroup$
    – M. Stern
    Oct 27 at 16:25

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