Result 279, Mathematical physics

Exact quantum factoring over a fixed finite gate set

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 formalization Algorithm or complexity result

The bigger picture

Why it matters

Factoring an integer means finding the primes that multiply to make it. The manuscript claims a quantum procedure that always succeeds, using resources that grow only polynomially with the number of input bits.

What changes?

The unreviewed manuscript reports a family of quantum circuits that returns the complete prime factorization, including repeated prime factors, of every integer N at least 2 with probability one. Circuit descriptions can be generated in polynomial time. Both the number of gates and the number of qubits have polynomial worst-case bounds in the input length. A single finite gate library works for all input sizes, with each gate acting on only a bounded number of qubits.

What does that help mathematicians do?

The claimed result would rule out the necessity of a nonzero failure probability for polynomial-resource quantum factoring in this circuit model. Its worst-case bounds also mean the guarantee is not restricted to typical inputs or a special class of integers. Researchers could thus distinguish the mathematical possibility of efficient, certain success from the separate challenge of making such circuits practical.

Are there practical applications?

The immediate value is foundational: the claim places exact factoring within a model using one finite gate library and polynomial resources, without sacrificing certainty. The abstract does not describe a hardware implementation or give concrete resource estimates. Polynomial scaling alone does not establish practical feasibility, so any real-world advantage would require further evidence about circuit costs and implementation.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Exact quantum factoring over a fixed finite gate set

September 25, 2026 24 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026,
  author = {{OpenAI}},
  title = {{Exact quantum factoring over a fixed finite gate set}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026/main.pdf}{OAI:Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/279.md.

Exact quantum factoring over a fixed finite gate set

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization constructs a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer N≥2N\ge2 with probability exactly one. The circuits use one fixed finite set of bounded-arity gates, and both the gate count and number of qubits have polynomial worst-case bounds in the bit length of NN. The probability-one conclusion is exact, rather than an asymptotic or bounded-error guarantee.

Comparator links

Result Comparator statement
Exact quantum factoring over a fixed finite gate set ExactQuantumFactoring.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.