Motivated by estimation of quantum noise models, we study the problem of learning a Pauli channel, or more generally the Pauli error rates of an arbitrary channel. By employing a novel reduction to the "Population Recovery" problem, we give an extremely simple algorithm that learns the Pauli error rates of an $n$-qubit channel to precision $\epsilon$ in $\ell_\infty$ using just $O(1/\epsilon^2) \log(n/\epsilon)$ applications of the channel. This is optimal up to the logarithmic factors. Our algorithm uses only unentangled state preparation and measurements, and the post-measurement classical runtime is just an $O(1/\epsilon)$ factor larger than the measurement data size. It is also impervious to a limited model of measurement noise where heralded measurement failures occur independently with probability $\le 1/4$.
We then consider the case where the noise channel is close to the identity, meaning that the no-error outcome occurs with probability $1-\eta$. In the regime of small $\eta$ we extend our algorithm to achieve multiplicative precision $1 \pm \epsilon$ (i.e., additive precision $\epsilon \eta$) using just $O\bigl(\frac{1}{\epsilon^2 \eta}\bigr) \log(n/\epsilon)$ applications of the channel.

中文翻译：

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通过种群恢复估计泡利误差

受量子噪声模型估计的启发，我们研究了泡利通道的学习问题，或者更一般地说是任意通道的泡利错误率。通过对“人口恢复”问题采用新颖的简化，我们给出了一个非常简单的算法，该算法仅使用 $O 就可以将 $n$-qubit 通道的泡利误差率学习到精确 $\ell_\infty$ 中的 $\epsilon$ (1/\epsilon^2) \log(n/\epsilon)$ 通道的应用。这是最佳的对数因子。我们的算法仅使用未纠缠状态准备和测量，并且测量后的经典运行时间只是比测量数据大小大一个 $O(1/\epsilon)$ 因子。它也不受测量噪声的有限模型的影响，其中预告的测量失败以概率 $\le 1/4 $ 独立发生。
然后我们考虑噪声通道接近身份的情况，这意味着无错误结果发生的概率为 $1-\eta$。在小 $\eta$ 的范围内，我们扩展我们的算法以仅使用 $O\bigl(\frac{1}{\epsilon ^2 \eta}\bigr) \log(n/\epsilon)$ 通道的应用。