When sensitive information is encoded in data, it is important to ensure the privacy of information when attempting to learn useful information from the data. There is a natural tradeoff whereby increasing privacy requirements may decrease the utility of a learning protocol. In the quantum setting of differential privacy, such tradeoffs between privacy and utility have so far remained largely unexplored. In this work, we study optimal privacy-utility tradeoffs for both generic and application-specific utility metrics when privacy is quantified by $(\varepsilon,δ)$-quantum local differential privacy. In the generic setting, we focus on optimizing fidelity and trace distance between the original state and the privatized state. We show that the depolarizing mechanism achieves the optimal utility for given privacy requirements. We then study the specific application of learning the expectation of an observable with respect to an input state when only given access to privatized states. We derive a lower bound on the number of samples of privatized data required to achieve a fixed accuracy guarantee with high probability. To prove this result, we employ existing lower bounds on private quantum hypothesis testing, thus showcasing the first operational use of them. We also devise private mechanisms that achieve optimal sample complexity with respect to the privacy parameters and accuracy parameters, demonstrating that utility can be significantly improved for specific tasks in contrast to the generic setting. In addition, we show that the number of samples required to privately learn observable expectation values scales as $Θ((\varepsilon β)^{-2})$, where $\varepsilon \in (0,1)$ is the privacy parameter and $β$ is the accuracy tolerance. We conclude by initiating the study of private classical shadows, which promise useful applications for private learning tasks.

翻译：当敏感信息被编码于数据中时，在尝试从数据中学习有用信息时确保信息的隐私性至关重要。存在一种自然的权衡关系：提高隐私要求可能会降低学习协议的效用。在量子差分隐私的背景下，此类隐私与效用之间的权衡迄今在很大程度上尚未得到探索。本文研究了当隐私通过 $(\varepsilon,\delta)$-量子局部差分隐私进行量化时，针对通用及特定应用效用度量的最优隐私-效用权衡。在通用设置中，我们聚焦于优化原始态与隐私化态之间的保真度和迹距离。我们证明，去极化机制在给定隐私要求下实现了最优效用。随后，我们研究了在仅能访问隐私化态的情况下，学习可观测量相对于输入态的期望值这一具体应用。我们推导了以高概率达到固定精度保证所需的隐私化数据样本数量的下界。为证明此结果，我们利用了现有关于私有量子假设检验的下界，从而首次展示了其操作层面的应用价值。我们还设计了在隐私参数和精度参数方面达到最优样本复杂度的私有机制，证明针对特定任务，相较于通用设置，效用可以得到显著提升。此外，我们表明，私有学习可观测量期望值所需的样本数量按 $Θ((\varepsilon \beta)^{-2})$ 缩放，其中 $\varepsilon \in (0,1)$ 为隐私参数，$\beta$ 为精度容限。最后，我们开启了私有经典阴影的研究，这有望为私有学习任务提供有价值的应用前景。