Sample Complexity Bounds for Scalar Parameter Estimation Under Quantum Differential Privacy

This letter presents tight upper and lower bounds for minimum number of samples (copies of a quantum state) required to attain a prescribed accuracy (measured by error variance) for scalar parameters estimation using unbiased estimators under quantum local differential privacy for qubits. Particularly, the best-case (optimal) scenario is considered by minimizing the sample complexity over all differentially-private channels; the worst-case channels can be arbitrarily uninformative and render the problem ill-defined. In the small privacy budget $\epsilon $ regime, i.e., $\epsilon \ll 1$ , the sample complexity scales as $\Theta \text {(}\epsilon ^{-2}$ ). This bound matches that of classical parameter estimation under local differential privacy. The lower bound however loosens in the large privacy budget regime, i.e., $\epsilon \gg 1$ . The upper bound for the minimum number of samples is generalized to qudits (with dimension d) resulting in sample complexity of $\mathcal {O}\text {(d}\epsilon ^{-2}$ ).