# A Geometric-Aware Perspective and Beyond: Hybrid Quantum-Classical Machine Learning Methods

arXiv: https://arxiv.org/abs/2504.06328 · HTML: https://mlsanity.com/paper/2504.06328/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 40 references that are in the map.
- First posted on arXiv 8 Apr 2025.

## Builds on (most cited first, 11 of 11)

- [Xception: Deep Learning with Depthwise Separable Convolutions](https://mlsanity.com/paper/1610.02357/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Quantum Natural Gradient](https://mlsanity.com/paper/1909.02108/) (2019)
- [Quantum-enhanced machine learning](https://mlsanity.com/paper/1610.08251/) (2016)
- [Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach](https://mlsanity.com/paper/1304.4344/) (2013)
- [Efficient Clustering on Riemannian Manifolds: A Kernelised Random Projection Approach](https://mlsanity.com/paper/1509.05536/) (2015)
- [Hybrid Classical-Quantum method for Diabetic Foot Ulcer Classification](https://mlsanity.com/paper/2110.02222/) (2021)

## Built on by (most cited first, 1 of 1)

- [Beyond the Expressivity-Trainability Paradox: A Dynamical Lie Algebra Perspective on Navigating Barren Plateaus in Quantum Machine Learning](https://mlsanity.com/paper/2606.31536/) (2026)

## Abstract (arXiv)

Geometric Machine Learning (GML) has shown that respecting non-Euclidean geometry in data spaces can significantly improve performance over naive Euclidean assumptions. In parallel, Quantum Machine Learning (QML) has emerged as a promising paradigm that leverages superposition, entanglement, and interference within quantum state manifolds for learning tasks. This paper offers a unifying perspective by casting QML as a specialized yet more expressive branch of GML. We argue that quantum states, whether pure or mixed, reside on curved manifolds (e.g., projective Hilbert spaces or density-operator manifolds), mirroring how covariance matrices inhabit the manifold of symmetric positive definite (SPD) matrices or how image sets occupy Grassmann manifolds. However, QML also benefits from purely quantum properties, such as entanglement-induced curvature, that can yield richer kernel structures and more nuanced data embeddings. We illustrate these ideas with published and newly discussed results, including hybrid classical -quantum pipelines for diabetic foot ulcer classification and structural health monitoring. Despite near-term hardware limitations that constrain purely quantum solutions, hybrid architectures already demonstrate tangible benefits by combining classical manifold-based feature extraction with quantum embeddings. We present a detailed mathematical treatment of the geometrical underpinnings of quantum states, emphasizing parallels to classical Riemannian geometry and manifold-based optimization. Finally, we outline open research challenges and future directions, including Quantum Large Language Models (LLMs), quantum reinforcement learning, and emerging hardware approaches, demonstrating how synergizing GML and QML principles can unlock the next generation of machine intelligence.