量子机器学习中无参考框架的泛化问题
No Reference-Free Generalization in Quantum Machine Learning
AI总结：
本文证明，在没有外部量子参考框架的监督学习中，若训练态未张成整个希尔伯特空间，所有正交于其张成空间的纯态将获得相同预测，揭示了量子机器学习泛化能力受限于参考信息缺失而非状态区分或优化能力。
摘要：
量子机器学习通常以量子系统指数级大的状态空间为动机，但这一承诺留下了一个基本的泛化问题：当训练数据没有提供优先基、测量框架或其他定向结构时，学习器如何为未见过的量子方向赋予不同含义？我们通过在没有外部量子参考框架的情况下制定监督学习来解决这个可识别性问题，使得预测不能依赖于希尔伯特空间坐标的任意选择。这一要求迫使学习到的分类器保持训练数据未破坏的所有酉对称性。我们证明，每当训练态未能张成整个希尔伯特空间时，所有正交于其张成空间的纯态必须获得相同的预测——即使这些态相互正交且一旦提供适当测量即可完美区分。因此，限制并非由状态区分、优化或计算能力引起，而是由缺失的参考信息所致。我们进一步在弱对称性破缺下建立了鲁棒版本，并表明在多量子比特系统上学习通用无结构概念需要指数级多的独立定向训练方向。数值示例展示了由此产生的预测坍缩及其受控松弛。我们的结果将特征映射、测量基、哈密顿量、局域性、对称性先验、架构和足够多样化的训练态识别为泛化的操作资源。核心含义是，希尔伯特空间维度本身并非可学习的特征空间：成功的量子机器学习必须指定赋予未见量子方向语义意义的物理结构。
英文摘要：
Quantum machine learning is often motivated by the exponentially large state space of quantum systems, but this promise leaves a basic generalization problem unresolved: how can a learner assign different meanings to unseen quantum directions when the training data provide no preferred basis, measurement frame, or other orienting structure? We address this identifiability problem by formulating supervised learning without an external quantum reference frame, so that predictions cannot depend on an arbitrary choice of Hilbert-space coordinates. This requirement forces the learned classifier to preserve every unitary symmetry left unbroken by the training data. We prove that whenever the training states fail to span the full Hilbert space, all pure states orthogonal to their span must receive the same prediction -- even when those states are mutually orthogonal and perfectly distinguishable once an appropriate measurement is supplied. The limitation is therefore not caused by state discrimination, optimization, or computational power, but by missing reference information. We further establish a robust version under weak symmetry breaking and show that learning generic unstructured concepts on multiqubit systems requires exponentially many independently oriented training directions. Numerical illustrations visualize the resulting prediction collapse and its controlled relaxation. Our results identify feature maps, measurement bases, Hamiltonians, locality, symmetry priors, architectures, and sufficiently diverse training states as operational resources for generalization. The central implication is that Hilbert-space dimension alone is not a learnable feature space: successful QML must specify the physical structure that gives unseen quantum directions semantic meaning.
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