量子机器学习缺乏参考系时,无法区分未知量子态,需引入物理结构赋予意义。
No Reference-Free Generalization in Quantum Machine Learning

- 在无外部参考系下设计量子学习模型,强制保持数据未打破的对称性。
- 当训练态不覆盖全希尔伯特空间时,正交补空间的纯态必得相同预测。
- 适用于需理解量子方向语义的任务,如多比特系统中概念泛化。
量子机器学习常以量子系统的指数级状态空间为动机,但这一优势带来一个基本泛化难题:当训练数据未提供优选基、测量框架或其他定向结构时,学习者如何为未见的量子方向赋予不同含义?本文通过构建无外部量子参考系的监督学习,使预测不依赖希尔伯特空间坐标的任意选择。该要求迫使学习器保留训练数据未打破的每个酉对称性。我们证明:只要训练态未张满整个希尔伯特空间,所有与之正交的纯态必须获得相同预测——即使这些态相互正交且可被合适测量完全区分。此限制并非源于态区分、优化或计算能力,而是缺失参考信息所致。我们进一步建立弱对称破缺下的鲁棒版本,并表明在多比特系统上学习通用无结构概念,需呈指数级独立定向的训练方向。数值实验可视化了预测坍缩及其受控松弛。结果揭示特征映射、测量基、哈密顿量、局域性、对称先验、架构及足够多样化的训练态是泛化操作资源。核心启示是:仅靠希尔伯特空间维度不足以构成可学习特征空间;成功的量子机器学习必须明确赋予未见量子方向语义的物理结构。
原文摘要 · Abstract (English)
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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