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：纠缠是赋予量子计算能力的关键资源。近期进展凸显了其对学习量子动力学的积极影响——将纠缠引入量子机器学习模型的量子操作或测量中，可显著减少训练数据规模，使其超越指定的预测误差阈值。然而，关于数据中纠缠程度如何影响模型性能的解析性理解仍不明确。本研究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定理，填补了这一认知空白。与先前结论相反，我们证明纠缠数据对预测误差的影响存在双重效应，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，或在保持相同预测误差下减少所需训练数据量；而当允许的测量次数有限时，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量子机器学习协议提供了关键指导，尤其适用于在量子资源受限的早期量子计算机上执行的方案。