Harnessing the potential computational advantage of quantum computers for machine learning tasks relies on the uploading of classical data onto quantum computers through what are commonly referred to as quantum encodings. The choice of such encodings may vary substantially from one task to another, and there exist only a few cases where structure has provided insight into their design and implementation, such as symmetry in geometric quantum learning. Here, we propose the perspective that category theory offers a natural mathematical framework for analyzing encodings that respect structure inherent in datasets and learning tasks. We illustrate this with pedagogical examples, which include geometric quantum machine learning, quantum metric learning, topological data analysis, and more. Moreover, our perspective provides a language in which to ask meaningful and mathematically precise questions for the design of quantum encodings and circuits for quantum machine learning tasks.

翻译：利用量子计算机在机器学习任务中的潜在计算优势，依赖于通过通常所称的量子编码将经典数据上传至量子计算机。此类编码的选择可能因任务而异，且仅有少数案例中，结构特性为其设计与实现提供了洞见，例如几何量子学习中的对称性。在此，我们提出一种观点：范畴论为分析保持数据集与学习任务内在结构的编码提供了一个自然的数学框架。我们通过教学示例阐释这一观点，包括几何量子机器学习、量子度量学习、拓扑数据分析等。此外，我们的视角提供了一种语言，用以提出有意义且数学精确的问题，从而指导量子机器学习任务中量子编码与电路的设计。