The Aaronson-Kuperberg unitary synthesis problem at constant diamond-norm error

Theorem 1.1: there are a polynomial $p$ and a deterministic algorithm outputting, on $1^n$ , an oracle circuit $A_n$ with at most $p(n)$ qubits, gates and oracle calls and oracle input length at most $p(n)$ , such that for every $U\in U(2^n)$ some $f$ gives $\|\Phi_{n,f}-\mathcal U\|_\diamond\le1/2$ . The method is a recursive dimension reduction using one coherently controlled smaller synthesis per step. It does not achieve arbitrarily small error, does not give an efficient classical procedure for the oracle, and answers positively a question on which Aaronson (2021) had conjectured a negative answer.

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Occurred: Oct 5, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Aaronson-Kuperberg unitary synthesis problem at constant diamond-norm error · Unitary synthesis, constant error

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VibeMathed record: The Aaronson-Kuperberg unitary synthesis problem at constant diamond-norm error

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No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against Aaronson's 2021 statement of Problem 6 (fetched for this check). Theorem 1.1 gives a circuit family generated in time polynomial in $n$ from $1^n$ (gates $H,T,T^\dagger,\mathrm{CNOT}$ ; polynomial qubits, gates, queries and query length) such that for every $U$ some Boolean oracle yields full diamond-norm error at most $1/2$ (no factor $1/2$ in the norm). The paper calls this the constant-error formulation; Aaronson's statement does not fix the error, and smaller-error versions are not addressed. The oracle is existential with no efficient construction. Not refereed here. No Lean formalization.

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The Aaronson-Kuperberg unitary synthesis problem at constant diamond-norm error

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Aaronson-Kuperberg 2007, Quantum versus classical proofs and advice

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Aaronson 2021, Open problems related to quantum query complexity (Problem 6)

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