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The use of geometric and symmetry techniques in quantum and classical information processing has a long tradition across the physical sciences as a means of theoretical discovery and applied problem solving. In the modern era, the emergent combination of such geometric and symmetry-based methods with quantum machine learning (QML) has provided a rich opportunity to contribute to solving a number of persistent challenges in fields such as QML parametrisation, quantum control, quantum unitary synthesis and quantum proof generation. In this thesis, we combine state-of-the-art machine learning methods with techniques from differential geometry and topology to address these challenges. We present a large-scale simulated dataset of open quantum systems to facilitate the development of quantum machine learning as a field. We demonstrate the use of deep learning greybox machine learning techniques for estimating approximate time-optimal unitary sequences as geodesics on subRiemannian symmetric space manifolds. Finally, we present novel techniques utilising Cartan decompositions and variational methods for analytically solving quantum control problems for certain classes of Riemannian symmetric space. Owing to its multidisciplinary nature, this work contains extensive supplementary background information in the form of Appendices. Each supplementary Appendix is tailored to provide additional background material in a relatively contained way for readers whom may be familiar with some, but not all, of these diverse scientific disciplines. The Appendices reproduce or paraphrase standard results in the literature with source material identified at the beginning of each Appendix. Proofs are omitted for brevity but can be found in the cited sources and other standard texts.