Quantum entanglement is a key resource for quantum computing, offering potential advantages over classical methods in various tasks like cryptography and optimization. The emerging field of quantum machine learning (QML) seeks to leverage these quantum advantages for machine learning tasks. Significant progress has been made in developing QML protocols demonstrating reduced complexity in learning quantum dynamics, often achieved by incorporating entanglement into quantum operations and measurements. However, the role of entanglement within the input data itself – entangled data – on QML performance remains largely unexplored. This study addresses this gap by investigating how the degree of entanglement in training data affects QML model performance in learning quantum dynamics. Prior work has hinted that highly entangled data might improve performance at the cost of increased resource requirements. This research challenges this assumption by establishing a quantum NFL theorem to rigorously analyze the impact of entangled data. The NFL theorem is a fundamental concept in machine learning, stating that no single learning algorithm universally outperforms others across all possible datasets. Understanding the quantum analog in the context of entangled data is crucial for designing efficient and effective QML algorithms, particularly for near-term quantum computers with limited resources.

Existing literature has demonstrated the benefits of entanglement in various aspects of quantum algorithms and quantum machine learning. Shor's algorithm, for instance, leverages entanglement to achieve polynomial-time factorization. Numerous quantum algorithms for optimization tasks have also shown speedups compared to classical counterparts. In the QML domain, various protocols incorporating entanglement in operations and measurements have shown promise in reducing complexity for specific learning tasks, particularly concerning learning quantum dynamics. These protocols often achieve advantages by incorporating entanglement into quantum operations and measurements, thereby reducing complexity. However, most studies overlook the impact of entanglement directly within the training data itself. Previous attempts at establishing quantum NFL theorems often relied on assumptions of infinite query complexity, failing to address the constraints of near-term quantum computing. This study builds upon these foundational works and addresses the crucial gap of understanding the role of entangled data in QML, particularly within the context of finite measurement limitations.

This research focuses on the task of incoherently learning quantum dynamics, where a quantum learner uses datasets with varying degrees of entanglement to infer the dynamics of an unknown unitary. The entanglement in the data is quantified by the Schmidt rank, *r*. The authors rigorously prove a quantum NFL theorem demonstrating the dual effect of entangled data on prediction error based on the number of allowed measurements, *m*. The learning process is modeled as follows: 1. **Entangled States:** The input states are entangled bipartite quantum states with a reference system, characterized by the Schmidt rank *r*. 2. **Incoherent Learning:** The quantum learner uses the entangled data to operate on an unknown unitary and infers its dynamics using finite measurement outcomes under projective measurement. 3. **Risk Function:** The performance is measured using a risk function, which is the average square error between the true output and the hypothesis output. The authors use the Haar unitary as the target unitary and construct a sampling rule for training input states that approximates a uniform distribution of entangled states with Schmidt rank *r*. 4. **Quantum NFL Theorem:** The authors prove a quantum NFL theorem, showing that the lower bound of the averaged prediction error depends on the number of measurements *m* and the Schmidt rank *r*. For sufficiently large *m*, increasing *r* reduces the prediction error or the required training data size. For small *m*, increasing *r* can increase the prediction error. 5. **Generalization:** The authors extend their analysis to a more general setting using arbitrary observables and positive-operator-valued measures (POVMs), demonstrating that the transition role of entangled data persists even under general measurements. 6. **Numerical Simulations:** Numerical simulations are performed to support the theoretical findings, showing that the prediction error decreases with increasing *m* and *r* when *m* is large enough, but increasing *r* for small *m* leads to increased error, confirming the transition role of entangled data.

The core finding is the establishment of a quantum NFL theorem showing the transition role of entangled data in learning quantum dynamics. This theorem reveals a dual effect of entanglement on prediction error, contingent upon the number of measurements allowed: * **Sufficient Measurements (*m* is large):** Increasing the Schmidt rank (*r*) of the entangled training data consistently reduces the prediction error, or equivalently, decreases the necessary training data size to achieve the same error. This aligns with the intuitive notion that more entangled data holds more information. * **Insufficient Measurements (*m* is small):** Increasing *r* can paradoxically *increase* the prediction error. This is because, with limited measurements, the learner struggles to extract the information contained within the highly entangled states. The increased complexity of highly entangled states outweighs the potential benefits of additional information. The authors further generalize this finding to scenarios involving arbitrary observables and POVMs, showing the transition role remains consistent. Increasing the number of possible outcomes in POVM significantly reduces the query complexity required for achieving the same prediction error. Numerical simulations confirm these theoretical results. The simulations show a clear transition point where increasing entanglement becomes detrimental to prediction accuracy when the number of measurements is limited. The simulations also demonstrate that increasing the training data size always leads to a better performance.

The findings challenge the prevailing assumption that entanglement in QML always leads to improved performance. The results emphasize the importance of balancing the information content of entangled data with the capacity to extract that information through measurements. The transition role of entanglement identified in this study deepens our understanding of the relationship between quantum information and QML. The theorem provides concrete guidance for designing QML algorithms, particularly those intended for near-term quantum computers. Knowing when and how to use entangled data efficiently is crucial for achieving optimal performance. The research highlights that utilizing highly entangled data might not always be advantageous and can be detrimental unless sufficient measurements are available to extract the relevant information. The optimal choice of entanglement degree is thus context-dependent and related to the available resources.

This paper presents a rigorous theoretical analysis of the impact of entangled data on quantum machine learning models for learning quantum dynamics. The authors establish a quantum NFL theorem demonstrating a transition role of entanglement, revealing a dual effect on prediction error depending on the number of measurements. The study provides valuable insights for designing QML algorithms for near-term quantum devices, highlighting the need to carefully consider the balance between the information content of entangled data and the measurement capabilities. Future research directions could explore the generalization of these findings to other QML tasks, investigate the combined effects of entangled data and entangled measurements, and develop practical strategies for optimizing the choice of entanglement level in various QML applications.

The study primarily focuses on the specific task of incoherently learning quantum dynamics under projective measurement. While the authors extend their analysis to general POVMs, exploring other QML tasks and learning protocols could further broaden the understanding of the transition role of entanglement. The numerical simulations, while supportive, are limited in scale due to computational constraints. Future studies with more extensive simulations could further refine our understanding of this phenomenon. The study uses Haar unitary as the target unitary, which might not perfectly represent all possible unitary operators.