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：我们引入了反向地图投影嵌入的新类别$(E_\alpha)_{\alpha \in [-\infty,1)}$，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。这些嵌入受到单位球面到其切平面的著名地图投影（实践中用于制图学）的启发，解决了幅度嵌入方法的常见缺陷——即数据点的标量倍数被等同看待，导致数据范数信息丢失。我们展示了反向地图投影如何作为量子机器学习中的等变嵌入使用。通过这些方法，我们可以利用经典数据集中的对称性来显著提升量子机器学习任务的性能。最后，我们选取四个$\alpha$值执行简单分类任务，以$E_\alpha$作为嵌入方式，并在等变与非等变两种设置下进行实验。我们将它们的实验结果与标准幅度嵌入的结果进行了对比。