Entanglement serves as the resource to empower quantum computing. Recent progress has highlighted its positive impact on learning quantum dynamics, wherein the integration of entanglement into quantum operations or measurements of quantum machine learning (QML) models leads to substantial reductions in training data size, surpassing a specified prediction error threshold. However, an analytical understanding of how the entanglement degree in data affects model performance remains elusive. In this study, we address this knowledge gap by establishing a quantum no-free-lunch (NFL) theorem for learning quantum dynamics using entangled data. Contrary to previous findings, we prove that the impact of entangled data on prediction error exhibits a dual effect, depending on the number of permitted measurements. With a sufficient number of measurements, increasing the entanglement of training data consistently reduces the prediction error or decreases the required size of the training data to achieve the same prediction error. Conversely, when few measurements are allowed, employing highly entangled data could lead to an increased prediction error. The achieved results provide critical guidance for designing advanced QML protocols, especially for those tailored for execution on early-stage quantum computers with limited access to quantum resources.

翻译：纠缠是赋能量子计算的资源。近期进展突显了其在学习量子动力学中的积极作用，将纠缠引入量子机器学习（QML）模型的量子操作或测量中，可显著减少超过指定预测误差阈值的训练数据规模。然而，数据中纠缠程度如何影响模型性能的分析性理解仍不明确。在本研究中，我们通过建立基于纠缠数据学习量子动力学的量子无免费午餐（NFL）定理来填补这一认知空白。与先前发现相反，我们证明纠缠数据对预测误差的影响具有双重效应，具体取决于允许的测量次数。当测量次数充足时，增加训练数据的纠缠度可持续降低预测误差，或减少在相同预测误差下所需的训练数据规模。反之，当允许的测量次数较少时，使用高纠缠数据反而可能导致预测误差增大。所得结果为设计先进的QML协议提供了关键指导，尤其是针对那些需在量子资源受限的早期量子计算机上运行的协议。