Eg for the qubit, the state space is the Bloch sphere, which is continuous.
Furthermore, quantum computing and classical computing are known to have the same computational power, just different computational complexity, ie. any quantum system can be simulated by a classical system with exponential slowdown, at worst.
There are plenty of papers exploring this territory [1,2,3] if you want further details. There are older references but they're not easily available online.
[1] Computability in Quantum Mechanics, 1995, https://philpapers.org/rec/MYRCIQ
[2] Effectively calculable quantum mechanics, 2015, https://arxiv.org/pdf/1508.03879.pdf
[3] Constructive physics, https://arxiv.org/pdf/0805.2859.pdf
But this is exactly my point. The fact that our observations have finite precision doesn't say anything about the underlying reality. That's the part that you're asserting. If it also turns out that assuming the underlying system is continuous results in much more compact theories than assuming otherwise, why is the obvious conclusion that spacetime is in fact discrete?
I'm not sure what you think I've asserted, but my only actual assertions are that all of modern physics requires only computable abstractions, and thus, continuous abstractions aren't strictly necessary.
I can certainly make the case that a continuous ontology requires more ontological commitments than a discrete ontology, but I haven't said anything along those lines thus far.
> If it also turns out that assuming the underlying system is continuous results in much more compact theories than assuming otherwise, why is the obvious conclusion that spacetime is in fact discrete?
What is the meaning of "compactness"? A proper comparison of theoretical parsimony requires comparing the Kolmogorov complexity. Compressing ontologies positing reals and/or rationals are definitely less parsimonious than ontologies positing only naturals.
As far as I can tell, by this argument there is literally nothing that would convince you that anything but a discrete theory was correct. So I don't think it's a very interesting argument, sorry. But thank you for explaining your viewpoint.
I do not see how that's true for quantum mechanics (take linear superposition as just one example). People may speculate about discrete underpinnings, but at this time, it's merely speculation.