I think you’re barking up the wrong tree. While I value Scott Aaronson’s words, unless he says “Grover’s algorithm scales this and this way”, almost no one in this forum will be able to evaluate what you’re saying. The discussion you’re trying to have belongs in another forum.
EDIT: okay, maybe in another thread. As I said, this one seems to come from a different position and should stick to the established knowledge.
He actually does talk about quantum search in that post, albeit in the limit case of cosmological scale. But believe it or not, I agree with you, so allow me to close with this, so as tie up some loose ends for the lurkers, to the extent that I can:
I would like to point out the Centauri Dreams post below, by Paul Gilster. (This blog is well-known in the astrophysics community, which is not to say that everything in it is generally accepted, just that Gilster is regarded as one of the more competent science journalists.) It discusses a 2004 experiment in which ESA’s Integral satellite observed a very powerful gamma ray burst, leading to the conclusion that any quantization of space must occur well below the Planck scale, and not greater than 10^(-48) m. I think this experiment deserved a lot more attention than it seems to have received. Indeed, it took quite a while for this to show up in my attempts to search the topic.
So one reason that I no longer expect QM to break down at the Planck scale is not because Dirac makes sense in integer spaces (which indeed it would not). It’s that, for all practical purposes, space is still continuous at the Planck scale according to the aforementioned experimental result (which I should probably accept even though there seems to be some dissent among scientists, and the original paper doesn’t seem to mention 10^(-48)). The other reason, as explained above, is that it sounds like Wolfram’s discrete metauniverse is indeed compatible with QM. I think that matters because we seem to know at least something is quantized at the Planck scale, and that would be entropy, if we take Aaronson at his word, so at some point we probably do need a unifying, quantized physical theory. (This is most likely how I ended up believing that space itself was quantized down there. I’ve heard the same meme from so many sources over so many years, and at some point must have just begun to assume that it was a done deal.)
It’s weird how we can have information quantization at one scale, and length quantization at who-knows-what scale. But from what I’ve read, that would seem to be the best current understanding.
I would also recommend reading the comments, which show how fervently some people are wedded to the idea of an analog world just because it’s more elegant, even though it’s just as untestable of a hypothesis as the string theory that most of them seem to hate. (This one guy points out that he uses pi in his physics equations, so therefore he uses real numbers, completely neglecting that it’s generated from a very small integer program.) At least, quantization of any sort at any scale is in principle refutable. (To be clear, I have no problem if integration (calculus, more generally) over a notional analog space is required to derive the quanta (units) of some physical quantity. This is how discrete Fourier transforms emerge from analog sine and cosine functions, for example. The problem I have is with the notion that any such quantity can be physically subdivided forever.)
https://www.centauri-dreams.org/2011/07/05/spacetime-beyond-the-planck-scale
Is it conceivable that QM could somehow still run out of ergodicity? Yes, and within certain limits, that would be testable. It’s just that, in light of the Integral observations, I now have no particular reason to assume that it happens at the Planck scale.
As to the QSN, it’s still unclear to me whether its empire waves and Fibbonacci chains have branching dynamics like Wolfram physics, which could allow it to keep pace with QM’s assumptions of discoverability, no matter how many qubits are involved. (While Wolfram et al have gone to great lengths to make their metatheory accessible from first principles, QGR unfortunately has not, and more to the point, doesn’t yet have a complete theory, although they’ve had some impressive success at deriving what are considered as arbitrary constants in other theories.)
And finally, to those who think that we’ll never have any hope of testing anything so small, have a look at what Fermilab has been up to: