Polynomial-time unitary synthesis from a Boolean oracle

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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.

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The paper states

Theorem 1.1 (Polynomial-time unitary synthesis)

In Polynomial-Time Unitary Synthesis from a Boolean Oracle, 5 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

There are a polynomial p and a deterministic classical algorithm which, on input 1^n for any integer n\ge1, outputs in time at most p(n) a quantum oracle circuit A_n with the following properties. The circuit uses the gates H,T,T^\dagger,\mathrm{CNOT} and a single Boolean oracle of input length m(n)\le p(n). Its total number of qubits, elementary gates, and oracle calls are each at most p(n). For every unitary U\in U(2^n) there is a function f:\{0,1\}^{m(n)}\to\{0,1\} for which the output channel \Phi_{n,f} of A_n^{O_f} satisfies

\norm{\Phi_{n,f}-\mathcal U}_\diamond\le\frac12.

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Polynomial-Time Unitary Synthesis from a Boolean Oracle

5 October 2026, current edition.

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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 \(H,T,T^\dagger,\mathrm{CNOT}\), 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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Yixin Ma, OpenAI mathematical physics, 25 families: a Chinese guided reading, 7 October 2026. Source
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17 references in Polynomial-Time Unitary Synthesis from a Boolean Oracle

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An unsigned public tier list, the OpenAI Math Tier List, classes it as a algorithm and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
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tparker, Philip Weiss, Will Sawin, Nik Weaver, Pace Nielsen, mathworker21, Joshua P. Swanson, Timothy Chow, Daniel Soltesz, MathOverflow: how many of the released results went against the conventional wisdom?, 10 October 2026. Source
A reply relays a complexity theorist’s remark that the positive unitary synthesis result is the opposite of what most experts expected.

tparker with nine answerers, How many of the results released by OpenAI went against the conventional mathematical wisdom?, 8 October 2026. Source
Chow notes the positive answer to unitary synthesis goes against what most of the field expected (quoting Aaronson).

From scottaaronson.blog, 7 October 2026. Source
Aaronson calls the positive answer to the unitary synthesis problem the opposite of what most experts expected and points to Dakshita Khurana’s account of working on it.

From dakshitakhurana.substack.com, 7 October 2026. Source
Links Dakshita Khurana’s blog post about their three years on unitary synthesis and their reaction to the AI proof.

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