Quantum entanglement is a crucial resource for learning properties from nature, but a precise characterization of its advantage can be challenging. In this Letter, we consider learning algorithms without entanglement to be those that only utilize states, measurements, and operations that are separable between the main system of interest and an ancillary system. Interestingly, we show that these algorithms are equivalent to those that apply quantum circuits on the main system interleaved with mid-circuit measurements and classical feedforward. Within this setting, we prove a tight lower bound for Pauli channel learning without entanglement that closes the gap between the best-known upper and lower bound. In particular, we show that $\mathrm{\Theta }(2^n{ϵ}^{-2})$ rounds of measurements are required to estimate each eigenvalue of an $n$-qubit Pauli channel to $ϵ$ error with high probability when learning without entanglement. In contrast, a learning algorithm with entanglement only needs $\mathrm{\Theta }({ϵ}^{-2})$ copies of the Pauli channel. The tight lower bound strengthens the foundation for an experimental demonstration of entanglement-enhanced advantages for Pauli noise characterization.

中文翻译：

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无纠缠泡利通道学习的严格界限

量子纠缠是从自然中学习特性的重要资源，但精确表征其优势可能具有挑战性。在这封信中，我们认为没有纠缠的学习算法是那些仅利用主要感兴趣系统和辅助系统之间可分离的状态、测量和操作的算法。有趣的是，我们证明这些算法相当于在主系统上应用量子电路并与中间电路测量和经典前馈交错的算法。在此设置中，我们证明了泡利通道学习的严格下界，没有纠缠，从而缩小了最著名的上限和下限之间的差距。特别是，我们表明$\mathrm{\theta }（2^n{\epsilon }^{-2}）$需要进行几轮测量来估计每个特征值$n$-量子比特泡利通道$\epsilon$在没有纠缠的情况下学习时，大概率会出现错误。相比之下，具有纠缠的学习算法只需要$\mathrm{\theta }（{\epsilon }^{-2}）$泡利频道的副本。严格的下限为泡利噪声表征的纠缠增强优势的实验演示奠定了基础。