Researchers have reported a rare case in which artificial intelligence has advanced one of the most abstract domains of mathematics — computational complexity theory. OpenAI’s GPT-5 suggested an idea that enabled the proof of strict limitations for the class of problems known as QMA, the quantum analogue of the well-studied NP class. In computer science, problems are classified according to how difficult they are to solve and to verify. The class NP contains problems for which finding a solution may be prohibitively hard, yet verifying a given solution can be done quickly. QMA extends this concept into the quantum realm. It is built upon the so-called Merlin–Arthur scheme, where Merlin acts as an all-powerful “prover,” sending a quantum state as evidence, while Arthur — armed only with a quantum verification algorithm — must decide whether to accept or reject it. Here, two parameters are crucial: completeness, the probability that a valid proof will be accepted, and soundness, the probability that a false proof will be rejected.
To reduce errors, researchers typically apply amplification methods: repeating the verification multiple times and aggregating the results. This is akin to cross-checking, where the consistency of several tests increases confidence. In 2025, Stacey Jeffery and Frik Witteveen demonstrated that completeness can approach unity at a double-exponential rate. Yet an open question remained: could this bound be pushed even further? Scott Aaronson of the University of Texas, together with Witteveen, pursued the problem but eventually reached an impasse. At that point, Aaronson decided to see whether GPT-5 could provide a useful insight. Its initial responses were incorrect, but after refinements, the model proposed framing the problem around a particular function that measures how closely the acceptance probability approaches absolute certainty. This insight proved decisive. By applying approximation techniques, the researchers showed that error amplification has an immutable ceiling: completeness cannot surpass a double-exponential closeness to one, and soundness cannot fall below an exponentially small threshold.
Thus, it became evident that black-box amplification methods in QMA have reached their limit. To answer the central question — whether QMA equals QMA1, where completeness is always exactly one — a fundamentally new approach will be required, one that probes not merely the external calls to the scheme but its inner structure. Writing on his blog, Aaronson remarked that he could now affirm artificial intelligence has entered what he once regarded as the most deeply “human” intellectual activity: proving distinctions between classes of quantum complexity.
Some colleagues dismissed GPT-5’s contribution as too obvious. Aaronson responded that indeed, it should have been obvious, and it would have been had researchers known the literature more thoroughly or invested more time in analysis. Despite the criticism, the work establishes the precise boundary of error amplification and closes a gap that had persisted in the theory for decades. The central question of whether QMA equals QMA1 remains unresolved, but the fact that AI has helped move the discussion forward marks a milestone: no longer is it merely about automating code or drafting papers, but about genuinely participating in the resolution of fundamental mathematical problems.