Quantum re-uploading models have been extensively investigated as a form of machine learning within the context of variational quantum algorithms. Their trainability and expressivity are not yet fully understood and are critical to their performance. In this work, we address trainability through the lens of the magnitude of the gradients of the cost function. We prove bounds for the differences between gradients of the better-studied data-less parameterized quantum circuits and re-uploading models. We coin the concept of $\textit{absorption witness}$ to quantify such difference. For the expressivity, we prove that quantum re-uploading models output functions with vanishing high-frequency components and upper-bounded derivatives with respect to data. As a consequence, such functions present limited sensitivity to fine details, which protects against overfitting. We performed numerical experiments extending the theoretical results to more relaxed and realistic conditions. Overall, future designs of quantum re-uploading models will benefit from the strengthened knowledge delivered by the uncovering of absorption witnesses and vanishing high frequencies.

中文翻译：

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量子重新上传模型的梯度和频率分布


量子重新上传模型作为变分量子算法背景下的一种机器学习形式已被广泛研究。它们的可训练性和表现力尚未完全理解，并且对它们的性能至关重要。在这项工作中，我们通过成本函数梯度的大小来解决可训练性问题。我们证明了经过更好的研究的无数据参数化量子电路和重新上传模型的梯度之间的差异的界限。我们创造了 $\textit{absorption witness}$ 的概念来量化这种差异。对于表达性，我们证明量子重新上传模型输出的函数具有消失的高频分量和相对于数据的上限导数。因此，此类函数对精细细节的敏感度有限，从而防止过拟合。我们进行了数值实验，将理论结果扩展到更轻松和更现实的条件下。总体而言，量子再上传模型的未来设计将受益于通过发现吸收证人和消失的高频而提供的强化知识。