Sample Complexity Bounds for Scalar Parameter Estimation Under Quantum Differential Privacy

Abstract

This paper 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 $\Theta(\epsilon^{-2})$. 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 $O(d\epsilon^{-2})$.