Quantum statistical queries provide a theoretical framework for investigating the computational power of a learner with limited quantum resources. This model is particularly relevant in the current context, where available quantum devices are subject to severe noise and have limited quantum memory. On the other hand, the framework of quantum differential privacy demonstrates that noise can, in some cases, benefit the computation, enhancing robustness and statistical security. In this work, we establish an equivalence between quantum statistical queries and quantum differential privacy in the local model, extending a celebrated classical result to the quantum setting. Furthermore, we derive strong data processing inequalities for the quantum relative entropy under local differential privacy and apply this result to the task of asymmetric hypothesis testing with restricted measurements. Finally, we consider the task of quantum multi-party computation under local differential privacy. As a proof of principle, we demonstrate that the parity function is efficiently learnable in this model, whereas the corresponding classical task requires exponentially many samples.

翻译：量子统计查询为研究受限量子资源的学习者的计算能力提供了一个理论框架。该模型在当前环境下尤为相关，因为可用的量子设备受到严重噪声影响且量子内存有限。另一方面，量子差分隐私框架表明，噪声在某些情况下能有利于计算，增强鲁棒性和统计安全性。在本工作中，我们建立了局部模型下量子统计查询与量子差分隐私之间的等价性，将经典领域中的著名结论扩展至量子场景。此外，我们推导了局部差分隐私下量子相对熵的强数据处理不等式，并将此结果应用于受限测量下的非对称假设检验任务。最后，我们研究了局部差分隐私下的量子多方计算任务。作为原理验证，我们证明了在该模型中奇偶校验函数是高效可学习的，而相应的经典任务需要指数级数量的样本。