Optimal Quantum Differential Privacy via Fisher Information Spectral Analysis

The Quantum Fisher Information (QFI) metric governs a fundamental duality: it quantifies both how precisely a parameter can be estimated (metrology) and how distinguishable two quantum states are (privacy). We exploit this duality to establish a geometry-aware framework for quantum differential privacy (DP) that replaces isotropic depolarizing noise with direction-dependent noise aligned to the QFI eigenstructure of the quantum embedding. We prove six principal theorems: (1) the minimax-optimal mechanism concentrates the noise budget in the dominant QFI eigenmode, achieving varepsilon = \(Δ²/2\)λ_max(1-cγ) with O\(d/λ_max\) advantage; (2) mixed-state QFI decomposition reveals that dephasing in the adversary's basis textit{increases} accessible information, while misaligned-basis dephasing provides constructive privacy amplification from hardware noise; (3) a tight privacy - utility uncertainty relation varepsilon cdot (1 - F) ge frac{Δ²}{2}frac{operatorname{Tr}(F)}{d}; (4) adaptive QFI estimation converging at O\(1/sqrt{n}\) yields 1.92times tighter bounds; (5) QFI-aligned composition saturates at O(1) versus O(k) for standard composition; and (6) hardware noise can be harnessed for privacy amplification. Adversarial vulnerabilities, Wasserstein guarantees, subspace projection, and a zero-knowledge audit protocol follow as corollaries. Results are validated on Qiskit Aer GPU simulations, IBM Quantum hardware ibm_fez, 156 qubits, and against classical DP baselines, achieving equivalent utility at varepsilon approx 0.001 versus varepsilon approx 4800 for classical DP.