A Lower Bound Framework for Quantum Functional Estimation

• URL: http://arxiv.org/abs/2608.02600v2
• Date: Mon, 10 Aug 2026 17:38:50 GMT
• Title: A Lower Bound Framework for Quantum Functional Estimation
• Abstract summary: We develop a unified framework for proving lower bounds for estimating functionals of $d$-dimensional quantum states.<n>The framework combines a Haar-random moment encoding process with moment matching and best approximation.
• Score: 13.491187998442596
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We develop a unified framework for proving lower bounds for estimating functionals of $d$-dimensional quantum states: \[ \mathcal L_φ(ρ)=\frac1d\sum_{i=1}^dφ(dλ_i(ρ)). \] The framework combines a Haar-random moment encoding process with moment matching and best polynomial approximation. A master theorem is provided to derive sample complexity lower bounds for estimating $\mathcal L_φ$ from the properties of $φ$. Using this framework, we resolve several open problems by establishing nearly tight lower bounds for a wide range of quantum property testing problems, including Uhlmann fidelity estimation, trace distance estimation, von Neumann/Rényi/Tsallis entropy estimation, spectrum estimation, and rank testing. Moreover, by quantum sample-to-query lifting, these sample lower bounds also imply quantum query lower bounds. These sample/query lower bounds imply the optimality of more than 20 quantum algorithms since 2015.

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