Simulation of non hermitian operators with Qiskit

Question

I am trying to simulate on a quantum computer a wavepaket evolution with a non unitary evolution operator (Hamiltonian with an absorbing (imaginary) potential for instance) and I found this post :
https://quantumcomputing.stackexchange.com/a/3928/21183
which contains exactly what I would like to do, that is applying $e^{iHt}e^{K \delta t}$ to a quantum state, where K is a tensor product of Pauli matrices. However, I don't quite understand how this can be performed. The post suggests that if we use an ancilla state $|\psi\rangle$ = $\alpha |0\rangle + \beta |1\rangle$ as a controlled-qubit to a controlled K-gate, then, if we measure the ancilla qubit in the $(\psi,\psi^\perp)$ basis and get $|\psi\rangle$, the target qubit will be in the state $(\cosh(\delta t) 1 +\sinh(\delta t) K)|q\rangle$, which is what we want. But I do not understand how this is possible: I have the impression that we can never measure the ancilla qubit in the state $|\psi\rangle$, since we would then get a non-unitary overall setup (target qubit + controlled qubit).
What am I getting wrong ? Would you have more precise references to simulate such systems on a QC ?
Thanks in advance for your help !