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

翻译：纠缠是学习中的一种有用资源，但精确刻画其优势具有挑战性。本研究将无纠缠学习算法定义为仅利用感兴趣主系统与辅助系统之间的可分态、测量及操作的算法。这类算法等价于在主系统上交替执行量子电路、中间电路测量与经典前馈的过程。我们证明了无纠缠学习泡利信道的紧下界，填补了已知最佳上界与下界之间的立方差距。特别地，我们表明：在无纠缠学习条件下，为以高概率估计n量子比特泡利信道的每个特征值至ε误差，需要Θ(2^n ε^{-2})轮测量。相比之下，有纠缠的学习算法仅需Θ(ε^{-2})轮测量。该紧下界为实验验证纠缠增强优势表征泡利噪声奠定了坚实基础。