They do match. The thing is that the description is much more complex, nothing else.

What I see from the preprint (notice that the authors have not discussed it with Quanta, because it is under peer-review) and bothers me is that they "add a real qubit" (...) which should be done very carefully because the two real qubits are not independent (they are linked by the hamiltonian).

I have not read the whole preprint, though.