Query-Optimal Estimation of Unitary Channels via Pauli Dimensionality
Sabee Grewal, Daniel Liang
TL;DR
This work introduces the concept of $k$-Pauli-dimensional unitary channels and provides a query-optimal tomography algorithm with complexity $O\left(\tfrac{2^k}{\varepsilon}\right)$, generalizing prior worst-case results (e.g., Haah et al.) to structured Pauli-spectrum subclasses. The authors develop Pauli projection as a key tool, enabling a convex combination of unitaries via a Pauli-twirl that can be implemented with block encodings using only forward access to the unknown unitary. Their approach reduces $k$-dimensional learning to approximately block-diagonal unitaries and then applies a block-diagonal tomography method with Pauli-projection-based rounding, followed by bootstrapping to Heisenberg scaling. This framework yields query-optimal algorithms for quantum $k$-juntas and for learning compositions of shallow circuits with near-Clifford circuits, providing exponential improvements in certain regimes and unifying prior restricted results. Altogether, the paper advances efficient quantum process tomography by exploiting Pauli-spectral structure, with potential practical impact on device characterization and quantum circuit learning.
Abstract
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.