{"ID":6138329,"CreatedAt":"2026-07-09T01:07:32.349475501Z","UpdatedAt":"2026-07-11T15:23:10.408442879Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.07540","arxiv_id":"2607.07540","title":"Towards Minimax Estimation of High-Order Functionals by Quantum Arguments","abstract":"We propose a novel approach to the minimax estimation of high-order functionals from the perspective of quantum computing. Specifically, for any real number $α\\gg 1$, we present two estimators, one for the classical functional $\\mathrm{F}_α(P) = \\sum_{i=1}^S p_i^α$ of a discrete distribution $P$ and the other for the quantum functional $\\mathrm{F}_α(ρ) = \\operatorname{tr}(ρ^α)$ of a mixed state $ρ$. These functionals have close connections with the Rényi entropy and the Tsallis entropy. We show that both estimators achieve the minimax optimal $L_2$ rate $α\\mathsf{n}^{-1}$ in the range $α\\lesssim \\mathsf{n} \\lesssim α^{3-o(1)}$, where the support size $S$ of $P$ or the dimension of $ρ$ can be much larger than the number of samples $\\mathsf{n}$. As a result, both estimators achieve the \\textit{optimal} sample complexity $\\mathsf{n} \\asymp α$, improving upon the prior best upper bounds $O(α^2)$ established by Jiao, Venkat, Han, and Weissman (IEEE Trans. Inf. Theory 2017) for classical functionals and Chen and Wang (COLT 2025) for quantum functionals. Our estimators are constructed under a unified framework using quantum primitives and run in linear time on a quantum computer. This work reveals an unexpected path from quantum computing to statistics, suggesting a conceptually new methodology for functional estimation. It adds to the growing list of quantum proofs for classical theorems.","short_abstract":"We propose a novel approach to the minimax estimation of high-order functionals from the perspective of quantum computing. Specifically, for any real number $α\\gg 1$, we present two estimators, one for the classical functional $\\mathrm{F}_α(P) = \\sum_{i=1}^S p_i^α$ of a discrete distribution $P$ and the other for the q...","url_abs":"https://arxiv.org/abs/2607.07540","url_pdf":"https://arxiv.org/pdf/2607.07540v1","authors":"[\"Qisheng Wang\"]","published":"2026-07-08T15:38:24Z","proceeding":"quant-ph","tasks":"[\"quant-ph\",\"cs.IT\",\"math.ST\"]","methods":"[]","has_code":false}