发表机构
Sharif University of Technology; University of Tehran; Iran University of Science and Technology（谢里夫理工大学; 德黑兰大学; 伊朗科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出层次Walsh-Fourier逼近框架用于变分量子分布学习，通过谱截断和热启动训练，给出端到端学习保证及总变差误差界，明确适用条件。
AI 中文摘要
我们通过布尔立方体上的Walsh-Fourier逼近层次来研究变分量子分布学习。在每个层次，选定的一组目标傅里叶系数定义了一个谱截断，该截断被投影到概率单纯形上，并用作量子电路Born机器的目标。在一个层次学习的参数通过热启动映射初始化下一个层次。我们证明了一个端到端的期望学习保证，其中逼近项由省略的傅里叶质量决定，而归一化的无偏估计器为经验截断提供了显式的统计界。然后，我们将总变差距离的抽象差异条件实例化，并将所得的分布误差与量子态保真度联系起来。总变差特化在我们的归一化$\ell_2$约定下产生显式因子$2^{n-1}$，因此仅对足够集中的傅里叶尾部具有信息量。该框架并未建立全局可训练性或消除贫瘠高原；相反，它确定了低到高谱训练允许逼近-估计-优化分析的条件。
英文摘要
We study variational quantum distribution learning through a hierarchy of Walsh--Fourier approximations on the Boolean cube. At each level, a selected set of target Fourier coefficients defines a spectral truncation, which is projected onto the probability simplex and used as the target of a quantum circuit Born machine. Parameters learned at one level initialize the next through a warm-start map. We prove an end-to-end expected learning guarantee where the approximation term is determined by the omitted Fourier mass, while a normalized unbiased estimator yields an explicit statistical bound for empirical truncations. We then instantiate the abstract discrepancy conditions for total variation distance and relate the resulting distributional error to quantum-state fidelity. The total-variation specialization incurs the explicit factor $2^{n-1}$ under our normalized $\ell_2$ convention and is therefore informative only for sufficiently concentrated Fourier tails. The framework does not establish global trainability or eliminate barren plateaus; rather, it identifies the conditions under which low-to-high spectral training admits a approximation--estimation--optimization analysis.
Comments18 pages, 3 figures
