Exact quantum factoring over a fixed finite gate set

What OpenAI claims

Gives a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer with probability one. Both gate count and qubit count are polynomial in the input length, and one fixed finite gate set suffices. (Lean)

Paper

1. Exact quantum factoring over a fixed finite gate set

   25 September 2026 · Preprint, not peer reviewed · PDF

   We give a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer N ≥ 2 with probability one. A fixed finite set of bounded-arity gates suffices, and both the gate count and the number of qubits have polynomial worst-case bounds in the input length.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.