That's my understanding as well. Quantum states encode finite information (despite their dynamics lying on the reals; the practical consequence is just that state transitions and information flow are smooth).

The formal discussion around this is largely centered on the Church-Turing thesis:

https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis

Essentially stating (in a certain interpretation) that any physical process can be simulated by a Turing machine (with no mention of efficiency). This seems to be the case, although I'm not sure we have a convincing proof from quantum field theory yet (note that quantum process can be simulated in classical Turing machines, although with an exponential cost).