Reverse Map Projections as Equivariant Quantum Embeddings

TL;DR

This paper introduces reverse map projection embeddings for quantum data encoding, inspired by cartographic map projections, which preserve data norms and leverage symmetries to improve quantum machine learning performance.

Contribution

It proposes a new class of quantum embeddings based on map projections, addressing amplitude embedding limitations and enhancing symmetry exploitation in quantum machine learning.

Findings

Reverse map projections preserve data norms better than amplitude embedding.

Using symmetries improves quantum classification accuracy.

Experimental results show advantages over standard amplitude embedding.

Abstract

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.