Reverse Map Projections as Equivariant Quantum Embeddings

TL;DR

This work introduces a new family of classical-to-quantum embeddings $E_α$ derived from higher-dimensional map projections to the unit sphere, addressing norm-information loss inherent to amplitude embedding. It proves that $E_α$ is equivariant for any unitary group representation by leveraging a corresponding direct-sum representation and a twirling projection, and demonstrates practical usefulness by embedding real-valued data into quantum states for a QML classification task. Four specific choices of $α$ (reverse gnomonic, stereographic, twilight, and orthographic) are evaluated against amplitude embedding on a $\,\mathbb{Z}_2\,$-symmetric boot-vs-sandal dataset, with both non-equivariant and equivariant QNN architectures. Across a range of data scalings, the equivariant models frequently outperform their non-equivariant counterparts, illustrating the potential for symmetry-exploiting embeddings to enhance quantum learning performance in the NISQ regime. Future work includes data-dependent selection of $α$, exploration of noise resilience, and deeper analysis of how embedding geometry interacts with circuit expressivity.

Abstract

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.

Figures (7)

• Figure 1: Observe that two right-angled triangles form in this diagram plotting $X_i$ against $X_n$, and derive Equation (\ref{['triangleeqn']}) from the ratio of their side lengths.
• Figure 2: In the case $n=1$ and $\alpha < -1$, the blue line segment is the set $R_\alpha$, embedded into two dimensional space as tangent to the unit circle. The set $S_\alpha$ is a subset of the unit circle and is depicted in green. The dashed lines are tangents to the unit circle crossing through the point $(\alpha,0)$.
• Figure 3: Footwear data from the fashion MNIST database xiao2017fashion. The task for the quantum neural network is to distinguish sandals (open footwear) vs boots (closed footwear). The images have been pixelated to contain $16\times 16 = 2^8$ pixels.
• Figure 4: Example data for the Reverse Twilight embedding illustrating - (a) That the final pixel was removed to make way for the additional coordinate from $E_\alpha$. (b) That this pixel is duplicated for the dataset in the equivariant case, so as to not serve as an indicator for which way the shoe is pointed.
• Figure 5: One repetition of the hardware-efficient ansatz for eight qubits.

Theorems & Definitions (20)

• Definition 3.1
• Remark 3.2
• Definition 3.3
• Lemma 3.4
• proof
• Proposition 3.5
• proof
• Definition 4.1
• Lemma 4.2
• proof