Understanding the power of quantum data in machine learning is central to many proposed applications of quantum technologies. While access to quantum data can offer exponential advantages for carefully designed learning tasks and often under strong assumptions on the data distribution, it remains an open question whether such advantages persist in less structured settings and under more realistic, naturally occurring distributions. Motivated by these practical concerns, we introduce a systematic framework based on quantum lossy data compression to bound the power of quantum data in the context of probably approximately correct (PAC) learning. Specifically, we provide lower bounds on the sample complexity of quantum learners for arbitrary functions when data is drawn from Zipf's distribution, a widely used model for the empirical distributions of real-world data. We also establish lower bounds on the size of quantum input data required to learn linear functions, thereby proving the optimality of previous positive results. Beyond learning theory, we show that our framework has applications in secure delegated quantum computation within the measurement-based quantum computation (MBQC) model. In particular, we constrain the amount of private information the server can infer, strengthening the security guarantees of the delegation protocol proposed in (Mantri et al., PRX, 2017).

翻译：理解量子数据在机器学习中的能力是许多量子技术应用方案的核心。尽管对于精心设计的学习任务，量子数据访问能够在强数据分布假设下提供指数级优势，但在结构较弱的环境和更现实的自然分布条件下，这种优势是否依然存在仍是未解之谜。基于这些实际问题，我们引入了一个基于量子有损数据压缩的系统性框架，用以界定量子数据在概率近似正确（PAC）学习背景下的能力边界。具体而言，我们针对从齐夫分布（一种广泛应用于现实世界数据经验分布的模型）中抽取数据的情形，给出了任意函数的量子学习器样本复杂度下界。同时，我们还建立了学习线性函数所需量子输入数据量的下界，从而证明了先前积极结果的最优性。在学习理论之外，我们展示了该框架在基于测量的量子计算（MBQC）模型中的安全委托量子计算方面的应用。特别地，我们约束了服务器可推断的私有信息量，从而增强了（Mantri等人，PRX，2017）所提出的委托协议的安全性保证。