I would first note that you can keep arguing with me all you want, but it is well known and commonly practiced that a Turing machine such as a digital computer can simulate a Quantum Computer or Quantum System to arbitrary precision, given sufficiently large but finite time. There are even cloud services offering such simulation capabilities, such as Amazon Braket [0]. The simulations take exponential time to run a linear time quantum algorithm, but they will produce the same result as a QC would (after enough sampling of the QC results).
> Hilbert space is infinite-dimensional and real-valued.
Infinite-dimensional Hilbert state spaces are a useful mathematical model, but they are not physical. Physical systems have finite elements and thus finite state spaces. We could argue that space itself is infinitely divisible, so that we need infinite-dimensional state spaces to represent the infinite possible positions of a particle, but this is not a physical concern, as it is impossible to differentiate in finite time two states that only differ in an infinitesimal position change - so, we are free to choose some minimal unit of length (say, one over Graham's number of the Planck length) and get a finite-dimensional state space. Crucially, for any amount of time and for any sensitivity of instruments, we can always choose some such unit and ensure that our results will not be distinguishable with those measurement instruments within that amount of time.
> QM says that there is no `rand(seed)`, rather there is only `rand()` that is what "no hidden variables" means.
QM says no such thing, though many interpretations do. Still, that is exactly why I chose rand(), not rand(seed) in my example, so I'm not sure why you're bringing this up. rand(seed) is a deterministic computation, rand() is a non-deterministic computation.
> If you want to do `rand(s) forall s`, that "forall" requires real numbers -- and then you're out-of-luck on computability.
This is not what I was proposing. I was proposing a Turing machine + perfectly random rand() as the simulation of a random world.
Even this is not necessary if you believe in the MWI, where the whole universe actually evolves perfectly linearly and deterministically from a fixed initial state, with no randomness of any kind. This interpretation is perfectly compatible with all observations of QM, if we also add the postulate that observers can only observe one state at a time (the one they are entangled with).
Even without going there, the construction I proposed earlier, where we essentially select enough real numbers to satisfy any possible measurement device leads to such that our computable set is indistinguishable in practice from the infinite uncomputable set we started with still applies.
In general, there is no (known) way to introduce infinity into empirical science - it is a priori impossible to distinguish, in finite time, between an arbitrarily large but finite quantity, and a truly infinite quantity.
[0] https://aws.amazon.com/braket/