Sample Complexity Bounds for Scalar Parameter Estimation Under Quantum Differential Privacy

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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 regime, i.e., 1, the sample complexity scales as (-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., 1. The upper bound for the minimum number of samples is generalized to qudits (with dimension d) resulting in sample complexity of O(d-2).

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