Query-Optimal Estimation of Unitary Channels via Pauli Dimensionality

We study process tomography of unitary channels whose Pauli spectrum is supported on a small subgroup. Given query access to an unknown unitary channel whose Pauli spectrum is supported on a subgroup of size $2^k$, our goal is to output a classical description that is $ε$-close to the unknown unitary in diamond distance. We present an algorithm that achieves this using $O(2^k/ε)$ queries, and we prove matching lower bounds, establishing query optimality of our algorithm. When $k = 2n$, so that the support is the full Pauli group, our result recovers the query-optimal $O(4^n/ε)$-query algorithm of Haah, Kothari, O'Donnell, and Tang [FOCS '23]. Our result has two notable consequences. First, we give a query-optimal $O(4^k/ε)$-query algorithm for learning quantum $k$-juntas -- unitary channels that act non-trivially on only $k$ of the $n$ qubits -- to accuracy $ε$ in diamond distance. This represents an exponential improvement in both query and time complexity over prior work. Second, we give a computationally efficient algorithm for learning compositions of depth-$O(\log \log n)$ circuits with near-Clifford circuits, where "near-Clifford" means a Clifford circuit augmented with at most $O(\log n)$ non-Clifford single-qubit gates. This unifies prior work, which could handle only constant-depth circuits or near-Clifford circuits, but not their composition.