Stabilizer Learning under Global Depolarizing Noise with Polynomial Copies and $2^t$ Time

Abstract

We study learning the stabilizer structure of an unknown pure $n$-qubit state observed through global depolarizing noise of unknown strength, with a fixed promised upper bound below one, using destructive two-copy Bell measurements. Let $0\le t\le n$ bound the stabilizer nullity, let $\varepsilon>0$ be the accuracy, and let $0<δ<1$ be the failure tolerance. With probability at least $1-δ$, a learner must return a commuting unsigned Pauli group containing every stabilizer of the pure state; averaging conjugation by that group must decrease the purity of the observed noisy state by at most $\varepsilon$. Tang and Steckmann gave such a learner with polynomially many copies and $2^t$ times polynomial time when the noise strength is below a threshold of at most about $0.16$, which tends to zero as $t/n\to1$. They asked whether other algorithms with these resources can exceed this threshold. We answer this question affirmatively for every fixed bound on global depolarizing noise below one and every $0\le t\le n$, including states with only one promised independent stabilizer. Our learner uses polynomially many copies and $2^t$ times polynomial classical running time. All resource bounds hold on every execution. Their polynomial factors depend on $n,1/\varepsilon,\log(1/δ)$, with degrees and constants depending on the fixed noise promise.