On OpenAI and Quantum Parallel Repetition

Some initial thoughts, and a complicated mix of feelings.

1. Wow. I mean, Erdos problems are cool (I genuinely mean that), I didn't know about the Jacobian conjecture before it got disproved. But this newest batch from OpenAI hits home in a way the previous announcements did not.

   New circuit lower bounds? A simple, easy-to-describe non-sofic group? Hardness of approximation for CVP without needing a unique games-like conjecture? I didn't just hear about these problems from my friends or from seminars. I feel their importance in my bones; I deeply care about the answers to these questions.

2. And then -- the kicker -- something that I personally spent a couple years on in grad school, leading to some of my proudest work: quantum parallel repetition theorems. I spent many hours, days, nights, weekends in cafes, in my office, at home, trying to understand Ran Raz's classical parallel repetition theorem and whether I could prove a quantum version of it.

   This period of struggle was important for me. I learned a lot of mathematics from it, and most importantly I learned that I could solve hard problems that (some) people cared about. I wouldn't say it's anywhere close to the most important problems in quantum information theory -- but nonetheless I fell in love with it as a grad student. I wouldn't want to trade that experience for anything.

3. Now, to the math. In 2016 I proved a polynomial-decay theorem; the exponential-decay theorem was left open and has since remained one of my favorite problems. I always intended to come back to it. Actually, a month ago I tried to set GPT 5.5 on it, and it didn't make very much progress.

   A couple days ago Lijie Chen sent me and a couple others a writeup. Life was busy so I didn't get a chance to look, but I guess now the cat's out of the bag so I probably should opine a bit.

   I presume the proof is correct (there's supposedly a Lean formalization, after all), but it will take me some time to digest it. I am gratified that it starts from where my paper left off, but goes beyond the limitations of my proof strategy by using some tricks and techniques that are probably known, in some collective fashion, to operator theorists and functional analysts. The reasoning document furnished by OpenAI is interesting but opaque: it states the problem, and then there's a leap of intuition on how to find the right purification using the "resolvent", and then does some exotic matrix entropy calculations to show that it works out.

   I don't understand it yet. Maybe it'll take me an afternoon to check all the calculations, but what would still be missing is why this was an approach that would've made sense in the first place. Is there some broader context or theory within which this would've been the obvious thing to do? What other results can be proven using these techniques? What is it telling us about quantum information or operator theory? I have no idea. I spent about an hour this morning asking ChatGPT these questions, but it's somewhat frustrating because it speaks with a mishmash of physicist, operator algebraist, quantum information theorist-lingo, plus the usual LLM breezy lilt that annoys everybody.

   I would love to hear from experts who find these calculations familiar. I might write about this again when I understand better.

4. I am disappointed by the writeup of this proof (sorry Lijie -- I should've taken a look at it earlier!). It writes in a way that's characteristic of a lot of ChatGPT-generated proofs, in which it elaborates at length on "boilerplate" setup, but then nonchalantly introduces what I consider to be the technical crux of the result: the particular Uhlmann transformation/dilation used to "align" all the states together. The technically interesting parts are buried deep into the paper, in Section 4, and introduced without any fuss or fanfare, as if this were the obvious thing to do.

   Normally, when I read a paper like this, I get immediately suspicious. What are they trying to hide by not explaining the new ideas up front and center? I wish OpenAI had spent a couple more prompts to clean up the writeup (I haven't had a chance to look at the other writeups so can't comment on their writing quality).

   Yes, there's a Lean proof. But that doesn't give me any understanding. That will just take time, I guess.

5. I'm in awe, and excited to see what other things we will learn from the AIs. There are a number of problems I've spent a long time thinking about, and maybe I will learn how to answer them soon.

6. But -- what then? What happens when AI has answered the handful of my favorite problems that I've spent the last 15 years thinking about? There are a lot of nice problems that I like, but it's not so easy to find a favorite problem.

7. There's a lot more to say here, but I think we mathematicians and theoreticians will have our work cut out for us: to keep a leash on these collossi of thought and reasoning, and to keep their abilities intelligible to humankind.