发表机构
Institute of Space Research German Aerospace Center (DLR)（德国航空航天中心空间研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究量子傅里叶模型中频率冗余的渐近分布，证明多种结构化特征值选择在层数增加时均趋于高斯分布，并揭示非结构化编码导致低频偏差，强调编码策略的重要性。
AI 中文摘要
文献中已表明，频谱的冗余分布会影响量子傅里叶模型（QFM）的表达能力和可训练性。在这项工作中，我们基于生成函数这一简单数学形式，探讨了数据重上传哈密顿量的特征值选择如何塑造该冗余频谱。我们针对每层相同的几种结构化特征值选择（包括等差数列和单量子比特泡利编码）推导了精确和渐近的冗余分布，并表明随着层数增加，两者均趋近于高斯分布。随后，我们证明这种高斯极限并非这些构造所特有，而是由每层相同的整数特征值构建的QFM的普遍特征。这些结果可与随机游走的中心极限定理相联系。这些结果阐明了为何通用或非结构化的编码选择会产生偏向低频的冗余偏差，凸显了在构建QFM时深思熟虑的编码策略的重要性。
英文摘要
The redundancy distribution of the frequency spectrum has been shown in the literature to impact the expressivity and trainability of Quantum Fourier Models (QFMs). In this work, we address the question of how this redundancy spectrum is shaped by the choice of eigenvalues of the data re-uploading Hamiltonians, using a simple mathematical formalism based on generating functions. We derive exact and asymptotic redundancy profiles for several structured eigenvalue choices identical for every layer, including arithmetic progressions and single-qubit Pauli encodings, and show that both approach a Gaussian profile as the number of layers grows. We then show that this Gaussian limit is not specific to these constructions but is a generic feature of QFMs built from integer eigenvalues that are identical in every layer. These results can be tied to the central limit theorem for random walks. These results clarify why generic or unstructured encoding choices give rise to a redundancy bias that favours low frequencies, highlighting the importance of well-thought-out encoding strategies when constructing a QFM.
Comments9 pages, 4 figures
