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It is incorrect to say that the time of simulation is $T O(\log N s^2t)$ (it's also better written as $O(T\log N s^2t)$). Conceptually, the simulation time ends at $O(\log N s^2t)$. If you want to consider the phase estimation, that's another (more complex) unitary, which includes hamiltonian simulation. If you were to think this in terms of function ...

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