Result 274, Mathematical physics

Parity is not in QAC0

Resolves Moore's parity conjecture in the measured-output model: constant-depth quantum circuits with arbitrary one-qubit gates, unbounded-arity Toffoli gates and polynomially many total qubits cannot compute parity with any fixed positive worst-case advantage. Ancillas start in zero, one output qubit is measured, and all other registers may be discarded. Xu–Li's reductions give the same bounded-error obstruction for strict majority.

Lean formalization Proof

The bigger picture

Why it matters

Parity asks whether a string of bits contains an odd number of ones. The manuscripts claim that even quantum circuits with gates acting on arbitrarily many qubits cannot reliably answer this question in constant depth under the stated resource limits.

What changes?

The unreviewed manuscripts report a barrier for circuits whose number of gate layers stays constant as inputs grow. Allowed gates are arbitrary single-qubit operations and Toffoli gates, which flip a target when all controls are on, with no limit on control count. With polynomially many total qubits, auxiliary qubits starting at zero, one measured output qubit, and unrestricted discarded registers, these circuits cannot compute parity with success probability exceeding one half by any fixed positive amount on every input.

What does that help mathematicians do?

The claimed obstruction does not require a circuit to leave its auxiliary qubits clean: it still applies when everything except the measured answer can be discarded. This rules out using unrestricted leftover quantum information to evade the bound. The summary also reports that Xu-Li's reductions transfer the bounded-error obstruction to strict majority, the task of deciding whether more than half the input bits are ones.

Are there practical applications?

The immediate value is foundational for quantum computation: the result identifies a limit on how much circuit depth can be compressed while keeping total qubit use polynomial in input length. It constrains parity and strict-majority computation in this particular gate model, rather than establishing a hardware benchmark or a limitation on quantum computing generally.

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.

2 manuscripts

Product-projection localization and the QAC0 parity lower bound

September 24, 2026 21 pages Main result formalized in Lean

We prove that constant-depth quantum circuits with arbitrary one-qubit and unbounded-arity Toffoli gates cannot compute parity with any fixed positive worst-case advantage using polynomially many qubits. Ancillas start in zero, only one output qubit is measured, and all final garbage is unrestricted. This resolves Moore's parity conjecture in the measured-output model.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026,
  author = {{OpenAI}},
  title = {{Product-projection localization and the $\mathrm{QAC}^0$ parity lower bound}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf}{OAI:Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026}},
  year = {2026}
}

Regular trajectories, pruning and quantum parity

September 24, 2026 27 pages Main result formalized in Lean

We prove that constant-depth quantum circuits with arbitrary one-qubit gates and unbounded-arity Toffoli gates cannot compute parity with any fixed positive worst-case advantage using polynomially many total qubits. Ancillary qubits are initialized to ∣0⟩|0\rangle, one output qubit is measured, and all other final registers may be discarded without restriction. This resolves Moore's parity conjecture in the measured-output model.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Regular-trajectories-pruning-and-quantum-parity-September-24-2026,
  author = {{OpenAI}},
  title = {{Regular trajectories, pruning and quantum parity}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026/paper.pdf}{OAI:Regular-trajectories-pruning-and-quantum-parity-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/274.md.

Parity is not in QAC0

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

Scope

The formalization rules out bounded-error parity computation by constant-depth quantum circuits with polynomially many zero-initialized ancillary qubits. For every fixed depth, polynomial bound on the total number of qubits, and 0<ε≤1/20<\varepsilon\le1/2, every sufficiently large input length has an input on which the measured-output parity success probability is less than 1/2+ε1/2+\varepsilon. A companion specialization gives success strictly below 2/32/3. The formalization also includes the product-projection localization estimate supporting this bound.

The formalized result supplies the polynomial-size parity consequence of the paper. For every fixed circuit depth and polynomial bound on the number of qubits, all sufficiently large input lengths have an input on which any such circuit computes measured-output parity with probability strictly below 2/32/3. Ancillary qubits start at zero. The formalization also contains the general positive-advantage parity bound and its product-projection localization estimate; regular-trajectory propagation and pruning are not separate formalized statements here.

Comparator links

Result Comparator statement
Constant-depth quantum parity lower bound QACParity.lean
Polynomial-size parity specialization RegularParity.lean
Polynomial-size quantum parity lower bound RegularParity.lean
General positive-advantage parity lower bound QACParity.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 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.