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

arXiv commentAccepted for publication in IEEE Control Systems Letters

References in corpus (9)

• RAPPOR: Randomized Aggregatable Privacy-Preserving Ordinal Response
• Generalized uncertainty relations: Theory, examples, and Lorentz invariance
• Fisher information under decoherence in Bloch representation
• Compatibility in Multiparameter Quantum Metrology
• Discontinuities of the quantum Fisher information and the Bures metric
• Quantum Pufferfish Privacy: A Flexible Privacy Framework for Quantum Systems
• Quantum metrology in the finite-sample regime
• Contraction of Private Quantum Channels and Private Quantum Hypothesis Testing
• Saturability of the Quantum Cramér-Rao Bound in Multiparameter Quantum Estimation at the Single-Copy Level