Sample Complexity Bounds for Scalar Parameter Estimation Under Quantum Differential Privacy

TL;DR

The paper investigates the minimum number of quantum state copies needed to estimate a scalar parameter under ε‑local quantum differential privacy. Using the Bloch‑sphere representation, quantum Fisher information, and the quantum Cramér–Rao bound, it derives tight upper and lower bounds on sample complexity for qubits, showing a quadratic dependence on the privacy budget in the small‑ε regime: $N_{\alpha,\epsilon}=\Theta(\alpha^{-1}\epsilon^{-2})$, with constants determined by the parameterization of the state. It also extends the upper bound to qudits, obtaining $O(d\epsilon^{-2})$, and shows that the quantum results align with classical local‑DP behavior in the small‑ε limit. The findings provide a precise link between privacy guarantees and data‑sample requirements in quantum privacy‑preserving parameter estimation, informing both theory and potential quantum‑privacy applications.

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 $ε$ regime, i.e., $ε\ll 1$, the sample complexity scales as $Θ(ε^{-2})$. 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., $ε\gg 1$. The upper bound for the minimum number of samples is generalized to qudits (with dimension $d$) resulting in sample complexity of $O(dε^{-2})$.