As Sebastian Riese points out, quantum mechanics is computable. Interestingly, classical mechanics is known to be non-computable. If classical mechanics were valid on all length and time scales, then you could construct a so-called "rapidly accelerating computer", which is a computer that accelerates such that the next clock cycle takes half the time to execute as its previous clock cycle. This means that an infinite amount of computations can be done in a finite time. One can then verify the truth of theorems and also verify whether that theorem then known to be true or false is actually provably true or false.
E.g. the Riemann hypothesis can be false, in which case it is provably false (just point of that zero that is not on the critical line), or it is true in which case there may or may not exist a proof for it. A proof is just an argument of finite length that demonstrates that it is true and such a proof may not exist.
The rapidly accelerating computer can simply check out all the zeros one by one and be done with the countably infinite number of zeros in a finite time and then return the result of whether or not they are all found to be on the critical line. Also, it can generate all proofs of theorems using Hilbert's proof checkers algorithm and then check if it ever encounters a proof of a theorem demonstrating that the Riemann hypothesis is true.
But of course, we know that classical mechanics is false. But while quantum mechanics is computable, this is only when you keep track of the unitary evolution of an isolated system. If you perform measurements, then in no-collapse interpretations, one assumes that all possible measurement outcomes are realized, and it's this entire set of measurement results that is computable. What is not computable are the individual results you observe in some particular sector. So, if you repeatedly measure the z-component of a spin polarized in the x-direction, you'll get a random set of measurement result. If spin down is replaced by 0 and spin up by 1, and you put a decimal point ( or is this called "binary point"?) in front then the number between 0 and 1 you get is non-computable.