(The quanta article is aware of the McKague paper. They cite it. I don't think I did anything that wasn't implied by that paper. In fact in the post I was less efficient; I needed 3 ancillary qubits per system instead of 1.)

(Ironically I got this from the references of the paper proposing the real-only test! They clearly understood the loophole while writing the paper. I wish they were clearer about it, and I think it's a fatal loophole in practice, but I no longer think they made a *mistake*.)

Another way to say it is...

Bell tests assume the verifier isn't correlated with the prover.

Real-only tests assume the prover isn't correlated with themselves.

One of these seems like a more ridiculous assumption, compared to the other.

The independence loophole applies to the verifier (which is me who I trust to be sufficient unpredictable) whereas the real-only loophole applies to the prover (Eve's computers).

It's a big difference. I know how Eve can build machines to spoof the real-only test; that's not true for Bell tests.

Apparently I was mistaken about linking to the playlist. The panel discussion is here: www.youtube.com/watch?v=f9VG...

For confused people checking the link and not seeing this anymore: the typo got fixed

The dialog takes >2n qubits of storage because we don't know any way to sift out the classically-known modulus (so annoying!). Still beats prior work.

There's some waste from GCDs taking so many steps, but the Bernstein-Yang GCD is close to the 2n qubit floor.

Because dialogs are built from linear operations, applying the dialog for x mod N to an initial state of (y, 0) instead of (x, 0) will produce (y/x, 0) instead of (1, 0). So y /= x is performed by:
1. convert x into dialog rep (i.e. do a GCD)
2. apply the dialog's steps to y
3. uncompute the dialog

I see loopholes

- CNOTs can't lower bound cost of computation because you can compile CNOTs away (e.g. Game of Surface Codes)
- What matters is amortized cost of doing many gates, not cost of isolated gates
- They speak of codes, not circuits, which is how Baspin+ missed the loophole in their bound

Someone pointed out they're saying "not entangled" in the abstract via the word "independent". I made a correction to the post.

I still think it's a silly assumption, and if I'd been writing the paper I'd have yelled it, but with hindsight I no longer think they hid it in the supplement.

I was careful to avoid saying it isn't in the paper. It isn't in the abstract/title/intro/conclusion/body of the paper; it's in the supplement of the paper.

I do think it's misleading to put this in the supplement. A paper isn't supposed to have tricksy fine print. Leave that to the lawyers.

This is tricky because any quantum algorithm, even one using imaginary values, can be encoded into the reals-only gates. They're BQP-complete. However, the simplest encodings add an ancilla that all encoded gates touch, and so fail to work in spacelike-separated tests that require locality.

Those papers misunderstand the goal. They avoid the word "complex" by using pairs of real numbers. But that's just complex with extra steps.

The real goal is to distinguish quantum computers limited to a reals-only gateset like

H, CCX, M, Z

from ones using a universal gateset like

H, CCX, M, ∜Z

This is the moral equivalent of running Bell tests without spacelike separation. It's better than nothing, but it's just fundamentally not nearly as convincing. There is a very clear avenue towards spoofing the test, and you would be blindly trusting that nature isn't doing it.