Varying the Hamiltonians between two fixed states Let us have a Hamiltonian $H_0$ and 2 states which can time evolve into each other via this Hamiltonian. In this particular situation, say one of the states evolves into the other in time $t_0$ .
Now let us fix these two states ; then we may have infinite Hamiltonians (other than $H_0$) that can time evolve one of them into the other. But the time taken for this evolution varies from $t_0$ and is say $t_H$ (that is, $t_H$ is the time Hamiltonian H takes to time evolve one of the fixed states into the other). From now on, we shall consider only those Hamiltonians that can time evolve one of the states into the other.
My question is given any small positive number $\epsilon$, will there exist a Hamiltonian H such that $t_H$ = $\epsilon$ ?
 A: As RoderickLee has commented already, the answer to your question as stated is yes. The reason is that rescaling any Hamiltonian $H$ with a dimensionless constant $\alpha$ speeds up the dynamics by a factor of $\alpha$.
But you might be interested to learn that quantum speed limits are an active area of research. The situation you are asking about is well understood already. The time $\tau$ that a quantum system takes to get from an initial state to an orthogonal final state satisfies the Mandelstam-Tamm bound
$$ \tau \geq \frac \pi 2 \frac \hbar {\Delta H} $$
and the Margolus-Levitin bound
$$ \tau \geq \frac \pi 2 \frac \hbar {\langle H \rangle} . $$
Technical notes: $H$ is normalized so that the ground state energy is zero. $\langle H \rangle$ and $\Delta H$ are the energy expectation value and its standard deviation, which are both constant during the evolution with the constant Hamiltonian $H$. For more info, see e.g. [Deffner 2017] (arXiv link).
These bounds show that, as long as the average energy and its fluctuations are bounded (thus, no rescaling allowed), you can not go arbitrarily fast from one state to another orthogonal state.
