Recent advances in classical machine learning have shown that creating models with inductive biases encoding the symmetries of a problem can greatly improve performance. Importation of these ideas, combined with an existing rich body of work at the nexus of quantum theory and symmetry, has given rise to the field of Geometric Quantum Machine Learning (GQML). Following the success of its classical counterpart, it is reasonable to expect that GQML will play a crucial role in developing problem-specific and quantum-aware models capable of achieving a computational advantage. Despite the simplicity of the main idea of GQML -- create architectures respecting the symmetries of the data -- its practical implementation requires a significant amount of knowledge of group representation theory. We present an introduction to representation theory tools from the optics of quantum learning, driven by key examples involving discrete and continuous groups. These examples are sewn together by an exposition outlining the formal capture of GQML symmetries via "label invariance under the action of a group representation", a brief (but rigorous) tour through finite and compact Lie group representation theory, a reexamination of ubiquitous tools like Haar integration and twirling, and an overview of some successful strategies for detecting symmetries.

翻译：经典机器学习的最新进展表明，构建具有问题对称性归纳偏置的模型能显著提升性能。将这些思想与量子理论与对称性交叉领域的丰富既有成果相结合，催生了几何量子机器学习这一新兴领域。鉴于其经典对应方法的成功，可以合理预期几何量子机器学习将在开发具备问题特异性和量子感知能力、并能实现计算优势的模型中发挥关键作用。尽管几何量子机器学习的核心思想——构建尊重数据对称性的架构——看似简单，其实践应用却需要大量的群表示理论知识。本文从量子学习视角出发，通过涉及离散群与连续群的关键示例，系统介绍表示理论工具。这些示例通过以下阐述而有机串联：以"群表示作用下的标签不变性"形式化捕获几何量子机器学习对称性，对有限群与紧致李群表示理论的简要（但严格）探讨，对哈尔积分与旋转等通用工具的重新审视，以及若干成功对称性检测策略的综述。