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.

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.