Anthropic's Internal Claude Model Solved an 18-Year Quantum Conjecture in Three Days

A non-public Anthropic model has produced a proof of the Aaronson-Ambainis conjecture, a question in quantum computing that has been open since 2008. The proof and its corollaries have been formalized in Lean, which means the result is machine-checkable rather than merely convincing.

Community observers suspect the model responsible is Claude Model 4, an internal version not available to the public.

Why the Result Matters

In 1994, Peter Shor published an algorithm that factors large integers in polynomial time, a task classical computers can only solve in exponential time. That result shook the foundations of modern cryptography and launched the quantum supremacy narrative — superposition, entanglement, exponential speedup, the sense that any sufficiently hard problem would fall once qubits were controlled.

The illusion that followed was that quantum computers could brute-force almost anything. Physicists and computer scientists corrected it: Shor's algorithm works because integer factorization hides strong periodic structure, and quantum machines are good at catching repetition. But if the input is genuinely random, with no structure to exploit, can a quantum computer still achieve exponential advantage?

In 2008, Scott Aaronson and Andris Ambainis reduced that physical question to pure mathematics. They treated reading an input as "observing" it: a classical machine reads an exact 0 or 1, while a quantum machine can spread one observation across a superposition of states. Their key observation was that the probability a quantum computer outputs "yes" after T observations can be characterized precisely as a bounded low-degree polynomial. If you can prove that such a polynomial must change substantially when some variable flips — that a variable with significant influence exists — then a classical computer can simulate the quantum machine at only polynomial extra cost.

That is the Aaronson-Ambainis conjecture. It now has a proof, which may soon make it the Aaronson-Ambainis theorem.

As the authors put it: quantum computing still has many open questions, but it now has one fewer.

Quantum advantage does not disappear with this result. What it removes is the blind belief that quantum computers beat classical ones everywhere; it draws the legal boundary of where quantum speedup can actually live.

The Harness, the Three Days, and the Lean Check

The work came out of Physical Superintelligence, which also resolved five long-open problems in quantum information theory last month. Earlier, researchers at MIT and Stanford had settled a special case of the classical-simulation corollary; they described the new proof as a major result with real implications for which problems can achieve quantum speedup.

The proof was not a literature lookup. It pulled together several strands of Boolean function analysis and probabilistic combinatorics in what the researchers called a rather remarkable way. Human specialists across several fields might eventually have gotten there, but not at this speed.

Once a harness built around Anthropic's internal model was standing, the first proof arrived in three days. It was then handed to the Lean proof assistant for verification. The intermediate results were subsequently reused on harder open problems, and in the process the exponents were tightened from 22 to 11 to 7 — a compressed picture of how discovery actually works: not a single flash of insight, but repeated sharpening toward the optimal bound.

Physics and mathematics remain vast, and the open questions outnumber the answered ones. What has changed is who is on the expedition. The team exploring the edges of the rational world is no longer composed only of humans.