# How to find minimum time needed for Hamiltonian evolution?

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 $$|t⟩$$.

• So do you mean $|s\rangle$ and $|t\rangle$ have the same eigenvalue? Oct 23 at 18:24
• @ZhiboYang Yes the eigen value is going to be same Oct 24 at 9:53
• What did you try so far? Did you calculate the unitary time evolution? Apply it to the initial state? Oct 24 at 12:10
• 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. Oct 27 at 3:34
• The usual series for the exponential function also works with matrices Oct 27 at 16:25