Theshani Nuradha Piliththuwasam Gallage

With sensitive information encoded in data, it is important to ensure the privacy to those sensitive information while learning useful information about the data. Towards that, quantum versions of differential privacy frameworks have been introduced. Privatizing data often comes with a cost. Meaning that, there are perfectly private mechanisms one could use that may lead to no utility depending on the application. However, privacy-utility tradeoffs in the quantum setting are not extensively studied. In this work, we study optimal privacy-utility tradeoffs for both generic and application-specific utility metrics when privacy is quantified by $(\varepsilon,\delta)$-quantum local differential privacy. As generic measures, we focus on optimizing fidelity and trace distance between the original state and the privatized state, showing that the depolarizing mechanism achieves the optimal utility while obtaining analytical expressions for utility for all privacy parameter regimes. Next, we study a specific application where one needs to learn the expectation of an observable with respect to an input state (property of quantum data), given access to only privatized states. There, we obtain a lower bound on the number of samples of privatized data required to achieve a fixed accuracy guarantee with high probability by utilizing lower bounds on private quantum hypothesis testing. We also obtain private mechanisms that achieve order optimality with respect to the privacy parameters and accuracy parameters, showcasing how the awareness of the task can be utilized to improve the utility in contrast to using mechanisms that optimize generic utility metrics. Furthermore, we show that the number of samples required to privately learn the expectation values scales as $\Theta((\varepsilon \beta)^{-2})$, where $\varepsilon \in (0,1)$ is the privacy parameter and $\beta$ is the accuracy tolerance. We also study a private version of classical shadows, which may be useful for the private estimation of properties of quantum states and processes.