We introduce the novel class $(E_\alpha)_{\alpha \in [-\infty,1)}$ of reverse map projection embeddings, each one defining a unique new method of encoding classical data into quantum states. Inspired by well-known map projections from the unit sphere onto its tangent planes, used in practice in cartography, these embeddings address the common drawback of the amplitude embedding method, wherein scalar multiples of data points are identified and information about the norm of data is lost. We show how reverse map projections can be utilised as equivariant embeddings for quantum machine learning. Using these methods, we can leverage symmetries in classical datasets to significantly strengthen performance on quantum machine learning tasks. Finally, we select four values of $\alpha$ with which to perform a simple classification task, taking $E_\alpha$ as the embedding and experimenting with both equivariant and non-equivariant setups. We compare their results alongside those of standard amplitude embedding.

翻译：我们引入了一类新的反向地图投影嵌入$(E_\alpha)_{\alpha \in [-\infty,1)}$，其中每个嵌入都定义了一种将经典数据编码到量子态中的独特新方法。这些嵌入的灵感来源于制图学中实际使用的、从单位球面到其切平面的著名地图投影，它们解决了振幅嵌入方法的一个常见缺陷——即数据点的标量倍数被等同对待，导致数据的范数信息丢失。我们展示了反向地图投影如何被用作量子机器学习中的等变嵌入。利用这些方法，我们可以利用经典数据集中的对称性，从而显著提升量子机器学习任务的性能。最后，我们选取了四个$\alpha$值，以$E_\alpha$作为嵌入，在等变与非等变两种设置下进行了一个简单的分类任务实验，并将它们的结果与标准振幅嵌入的结果进行了比较。