Let's assume that you have a Hermitian matrix
$$
H=\left(\begin{array}{cc} 0 & A^\dagger \\ A & 0 \end{array}\right).
$$
Let $|b\rangle$ be the state that we want to apply $A$ to, extended to work on the space that $H$ acts on. So, our aim is to implement $H|b\rangle$.
Let $X$ be the standard Pauli $X$ matrix. If we implement a unitary evolution
$$
U=e^{-i X\otimes H\delta t},
$$
then the action of this on an input state $|0\rangle|b\rangle$ can be expressed through a Taylor expansion, assuming $\delta t\|H\|\ll 1$:
$$
U|0\rangle|b\rangle=|0\rangle|b\rangle-i\delta t |1\rangle(H|b\rangle)-\frac{\delta t^2}{2}|0\rangle(H^2|b\rangle)+\ldots.
$$
So, if we measure the ancilla qubit in the standard basis, then with probability $\delta t^2\|H|b\rangle\|^2$, we find the ancilla to be in state $|1\rangle$, and the other register is in the desired state $H|b\rangle$, up to an accuracy $O(\delta t^2)$ (coming from the $O(\delta t^4)$ term in the Taylor expansion). If you don't get the $|1\rangle$ answer to your measurement, then to accuracy $O(\delta t^2)$, you've stil got the original state $|0\rangle|b\rangle$, so you can just repeat until success.
By that simple method, the overall error is very bad - you have to repeat $O(1/\delta t^2)$ times so that the overall error becomes $O(1)$. I'm sure this could be improved by replacing the $X$ with a cycle operation on a larger dimensional Hilbert space. However, note that this method is very similar to part of the HHL algorithm (where they do the controlled-rotation from the phase estimation register onto an ancilla) and, at least there, it is claimed that it can be improved using amplitude amplification. So I assume (without having looked at the details) that something similar is possible here.