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:
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).
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).