Reverse Map Projections as Equivariant Quantum Embeddings
This work tackles the problem of information loss in quantum data embeddings for researchers in quantum machine learning, representing an incremental improvement over existing techniques.
The authors introduced reverse map projection embeddings to encode classical data into quantum states, addressing the loss of norm information in amplitude embeddings, and demonstrated improved performance in a classification task compared to standard methods.
We introduce the novel class $(E_α)_{α\in [-\infty,1)}$ 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 $α$ with which to perform a simple classification task, taking $E_α$ as the embedding and experimenting with both equivariant and non-equivariant setups. We compare their results alongside those of standard amplitude embedding.