Towards Minimax Estimation of High-Order Functionals by Quantum Arguments

2026-07-08

AI Summary

This study addresses the minimax estimation of high-order functionals—such as Rényi and Tsallis entropies—in regimes where the sample size is far smaller than the support size of the distribution or the dimension of the quantum state. It introduces quantum primitives for the first time to construct a unified framework encompassing both classical and quantum estimation, achieving the $L_2$-optimal convergence rate in linear time on a quantum computer. Within the range $\alpha \lesssim n \lesssim \alpha^{3 - o(1)}$, the estimator attains the optimal rate $\alpha/n$, dramatically reducing the required sample complexity from the previous $O(\alpha^2)$ to nearly linear scaling $n \sim \alpha$. Beyond establishing minimax optimality, this work pioneers a new paradigm of quantum-enhanced statistical inference and provides quantum-inspired proofs for classical statistical problems.

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.

Problem

• minimax estimation
• high-order functionals
• Rényi entropy
• quantum state
• sample complexity

Innovation

• minimax estimation
• quantum computing
• high-order functionals
• sample complexity
• Rényi entropy