Result 283, Mathematical physics

Polynomial-time unitary synthesis from a Boolean oracle

Solves the constant-error Aaronson–Kuperberg unitary synthesis problem: a uniform polynomial-size quantum oracle circuit approximates every n-qubit unitary channel within diamond-norm error 1/2, after a suitable Boolean oracle is chosen. Gates, qubits, oracle calls and query length are polynomially bounded. The target-dependent oracle may have an unrestricted truth table; its efficient classical construction is not asserted.

Proof

The bigger picture

Why it matters

The manuscript reports that a compact quantum circuit can approximate any reversible quantum transformation if supplied with the right Boolean oracle. The key distinction is between efficiently using that black box and efficiently building it.

What changes?

For every number n of qubits, the manuscript reports a circuit generated in polynomial time from n alone. Choosing a target-dependent Boolean oracle, a black box evaluating a function with a one-bit output, lets it approximate any n-qubit unitary channel, the action of a reversible quantum transformation on states. The full diamond-norm error is at most 1/2, including inputs entangled with other systems. Qubits, gates, oracle calls and query length all have polynomial bounds in n.

What does that help mathematicians do?

The circuit uses only H, T, inverse T and CNOT as elementary gates, alongside oracle access. The claimed result therefore rules out a circuit-size obstruction at this fixed error tolerance in this oracle model. One circuit structure serves every target of a given size; only the oracle changes. This separates the cost of quantum processing from the target information supplied through the oracle.

Are there practical applications?

Its immediate value is foundational for quantum computation: it clarifies what access to target-dependent Boolean information can enable. It does not establish an efficient compiler for arbitrary quantum transformations. The oracle's truth table is unrestricted, and efficient classical construction is not asserted. Nor does the stated constant-error result establish synthesis at arbitrarily high accuracy.

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

Polynomial-Time Unitary Synthesis from a Boolean Oracle

October 5, 2026 21 pages

We give a positive answer to the constant-error formulation of the Aaronson–Kuperberg unitary synthesis problem. For every n, a quantum oracle circuit generated in polynomial time from n alone can approximate the channel of every n-qubit unitary to full diamond-norm error at most 1/2, after a suitable Boolean oracle is chosen. Using the fixed gates H,T,T†,CNOTH,T,T^\dagger,\mathrm{CNOT}, the circuit has polynomially many qubits, elementary gates, and oracle calls, and its oracle queries have polynomial length. The oracle may depend on the target unitary; the theorem does not give an efficient classical procedure for constructing it.

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Cite (BibTeX) 
@misc{OAI:Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026,
  author = {{OpenAI}},
  title = {{Polynomial-Time Unitary Synthesis from a Boolean Oracle}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf}{OAI:Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026}},
  year = {2026}
}

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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.