Reverse Map Projections as Equivariant Quantum Embeddings

• URL: http://arxiv.org/abs/2407.19906v2
• Date: Mon, 19 Aug 2024 09:40:32 GMT
• Title: Reverse Map Projections as Equivariant Quantum Embeddings
• Authors: Max Arnott, Dimitri Papaioannou, Kieran McDowall, Phalgun Lolur, Bambordé Baldé,
• Abstract summary: We introduce the novel class $(E_alpha)_alpha in [-infty,1)$ of reverse map projection embeddings. Inspired by well-known map projections from the unit sphere onto its tangent planes, these embeddings address the common drawback of the amplitude embedding method. We show how reverse map projections can be utilised as equivariant embeddings for quantum machine learning.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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