Full Scott Aaronson quote in case anyone else is interested:
This is the first paper I’ve ever put out for which a key technical step in the proof of the main result came from AI—specifically, from GPT5-Thinking. Here was the situation: we had an N×N Hermitian matrix E(θ) (where, say, N=2n), each of whose entries was a poly(n)-degree trigonometric polynomial in a real parameter θ. We needed to study the largest eigenvalue of E(θ), as θ varied from 0 to 1, to show that this λmax(E(θ)) couldn’t start out close to 0 but then spend a long time “hanging out” ridiculously close to 1, like 1/exp(exp(exp(n))) close for example.
Given a week or two to try out ideas and search the literature, I’m pretty sure that Freek and I could’ve solved this problem ourselves. Instead, though, I simply asked GPT5-Thinking. After five minutes, it gave me something confident, plausible-looking, and (I could tell) wrong. But rather than laughing at the silly AI like a skeptic might do, I told GPT5 how I knew it was wrong. It thought some more, apologized, and tried again, and gave me something better. So it went for a few iterations, much like interacting with a grad student or colleague. Within a half hour, it had suggested to look at the function
(the expression doesn’t copy-paste properly)
It pointed out, correctly, that this was a rational function in θ of controllable degree, that happened to encode the relevant information about how close the largest eigenvalue λmax(E(θ)) is to 1. And this … worked, as we could easily check ourselves with no AI assistance. And I mean, maybe GPT5 had seen this or a similar construction somewhere in its training data. But there’s not the slightest doubt that, if a student had given it to me, I would’ve called it clever. Obvious with hindsight, but many such ideas are.
I had tried similar problems a year ago, with the then-new GPT reasoning models, but I didn’t get results that were nearly as good. Now, in September 2025, I’m here to tell you that AI has finally come for what my experience tells me is the most quintessentially human of all human intellectual activities: namely, proving oracle separations between quantum complexity classes.
Full Scott Aaronson quote in case anyone else is interested:
(couldn’t resist including that last sentence)