Sample Complexity Bounds for Scalar Parameter Estimation Under Quantum Differential Privacy

TL;DR

This paper derives tight bounds on the number of quantum state copies needed for accurate scalar parameter estimation under quantum local differential privacy, revealing the dependence on privacy budget and system dimension.

Contribution

It provides the first tight bounds on sample complexity for quantum parameter estimation under local differential privacy, including extensions to qudits.

Findings

Sample complexity scales as Θ(ε^{-2}) for small privacy budgets.

Upper bounds extend to qudits with complexity O(dε^{-2}).

Bounds match classical estimation in the small privacy regime.

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