Information-Theoretic Lower Bounds for Approximating Monomials via Optimal Quantum Tsallis Entropy Estimation

2025-09-03

AI Summary

This work establishes a novel connection between information theory and approximation theory, addressing the fundamental problem of deriving information-theoretic lower bounds on the polynomial approximation error of monomials $x^n$. To this end, we introduce the first integration of quantum Tsallis entropy estimation with polynomial approximation, leveraging the quantum oracle model and optimal sampling analysis to obtain an $Omega(sqrt{n})$ information-theoretic lower bound on the approximation degree. Furthermore, we construct the first quantum Tsallis entropy estimator achieving optimal query complexity—up to logarithmic factors—across all parameter regimes; specifically, we improve the query complexity from $O(1/varepsilon)$ to $widetilde{O}(1/(sqrt{q},varepsilon))$, attaining optimality for integer-order Tsallis entropy estimation. This work not only extends the theoretical frontiers of quantum entropy estimation but also provides new analytical tools for approximation problems in both classical and quantum information theory.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

This paper reveals a conceptually new connection from information theory to approximation theory via quantum algorithms for entropy estimation. Specifically, we provide an information-theoretic lower bound $Ω(sqrt{n})$ on the approximate degree of the monomial $x^n$, compared to the analytic lower bounds shown in Newman and Rivlin (Aequ. Math. 1976) via Fourier analysis and in Sachdeva and Vishnoi (Found. Trends Theor. Comput. Sci. 2014) via the Markov brothers' inequality. This is done by relating the polynomial approximation of monomials to quantum Tsallis entropy estimation. This further implies a quantum algorithm that estimates to within additive error $varepsilon$ the Tsallis entropy of integer order $q geq 2$ of an unknown probability distribution $p$ or an unknown quantum state $ρ$, using $widetilde Θ(frac{1}{sqrt{q}varepsilon})$ queries to the quantum oracle that produces a sample from $p$ or prepares a copy of $ρ$, improving the prior best $O(frac{1}{varepsilon})$ via the Shift test due to Ekert, Alves, Oi, Horodecki, Horodecki and Kwek (Phys. Rev. Lett. 2002). To the best of our knowledge, this is the first quantum entropy estimator with optimal query complexity (up to polylogarithmic factors) for all parameters simultaneously.

Problem

• Estimate quantum Tsallis entropy with optimal query complexity
• Provide information-theoretic lower bounds for monomial approximation
• Connect information theory to approximation theory via quantum algorithms

Innovation

• Quantum Tsallis entropy estimation method
• Optimal query complexity algorithm
• Information-theoretic lower bounds approach