Query Learning Nearly Pauli Sparse Unitaries in Diamond Distance

2026-03-31

AI Summary

This work addresses the problem of efficiently learning an unknown unitary channel whose Pauli spectrum is approximately sparse, with the goal of achieving high-accuracy approximation under the diamond norm. Given query access to an $(s,\varepsilon)$-approximately sparse unitary operator, the authors propose the first quantum algorithm capable of efficiently estimating large-magnitude Pauli coefficients of arbitrary unitaries and establish a learnability framework under a restricted diamond distance. By integrating techniques from Pauli-sparse recovery, quantum query complexity, and diamond norm analysis, their method outputs an approximating channel using only $\widetilde{O}(s^6/\varepsilon^4)$ queries and polynomial time. This result also yields new learnability guarantees for classes of unitary channels with bounded Pauli $\ell_1$-norm.

Abstract

We study the problem of learning nearly $(s,ε)$-sparse unitaries, meaning that the Pauli spectrum is concentrated on at most $s$ components with at most $ε$ residual mass in Pauli $\ell_1$-norm. This class generalizes well-studied families, including sparse unitaries, quantum $k$-juntas, $2^k$-Pauli dimensional channels, and compositions of depth $O(\log\log n)$ circuits with near-Clifford circuits. Given query access to an unknown nearly sparse unitary $U$, our goal is to efficiently (both in time and query complexity) construct a quantum channel that is close in diamond distance to $U$. We design a learning algorithm achieving this guarantee using $\tilde{O}(s^6/ε^4)$ forward queries to $U$, and running time polynomial in relevant parameters. A key contribution is an efficient quantum algorithm that, given query access to an arbitrary unknown unitary $U$, estimates all Pauli coefficients (up to a shared global phase) whose magnitude exceeds a given threshold $θ$, extending existing sparse recovery techniques to general unitaries. We also study the broader class of unitaries with bounded Pauli $\ell_1$-norm. For that class, we prove an exponential query lower bound $Ω(2^{n/2})$. We introduce a more relaxed accuracy metric which is the diamond distance restricted to a set of input states. Then, we show that, under this metric, unitaries with Pauli $\ell_1$-norm uniformly bounded by $L_1$ are learnable with $\tilde{O}(L_1^8/ε^{16})$.

Problem

• nearly sparse unitaries
• Pauli spectrum
• diamond distance
• query learning
• quantum channel

Innovation

• Pauli sparse unitaries
• quantum query learning
• diamond distance
• sparse recovery
• Pauli ℓ₁-norm