用范畴论设计保持数据结构的量子编码，提升量子机器学习效果
Towards structure-preserving quantum encodings
- 以范畴论为工具，构建保持数据几何与拓扑结构的量子编码方法
- 在几何学习、度量学习和拓扑数据分析中验证了编码有效性
- 为量子机器学习编码设计提供数学严谨的新视角，适合理论研究者
将经典数据映射到量子计算机是实现量子计算在机器学习中潜力的关键步骤，这一过程依赖于所谓的量子编码。编码的选择因任务而异，目前仅有少数情况（如几何量子学习中的对称性）利用结构指导其设计。本文提出范畴论可作为分析保留数据与学习任务内在结构的量子编码的自然数学框架。通过教学示例，涵盖几何量子机器学习、量子度量学习、拓扑数据分析等场景，展示了该框架的有效性。此外，该视角为量子机器学习编码与电路的设计提供了精确的数学语言，能提出有意义且形式严谨的问题。
原文摘要 · Abstract (English)
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.
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