Researchers Link Information Theory to Approximation, Estimating Tsallis Entropy with Algorithms Using Sample Queries !

 

The challenge of accurately representing complex mathematical functions lies at the heart of numerous scientific fields, and researchers continually seek ways to improve approximation techniques. Qisheng Wang from the University of Edinburgh, alongside colleagues, now demonstrates a fundamental link between information theory and polynomial approximation, revealing inherent limits to how well we can represent basic building blocks of mathematics called monomials. This work establishes a theoretical lower bound on the accuracy of these approximations, surpassing previous analytical methods, and crucially, leads to a new algorithm for estimating Tsallis entropy, a measure of uncertainty in probability distributions or quantum states. The team’s algorithm achieves optimal efficiency, requiring the fewest possible measurements to determine entropy for a wide range of parameters, representing a significant advance in the field of quantum information science and statistical estimation.

Researchers build upon previous work, utilising established mathematical principles to connect polynomial approximation of monomials with quantum Tsallis entropy estimation. This connection yields a quantum algorithm capable of estimating the Tsallis entropy of integer order for an unknown probability distribution or quantum state, achieving an additive error. The algorithm requires fewer queries to a quantum oracle than previous methods, representing an improvement over the Shift test. To the best of the researchers’ knowledge, this represents the first quantum entropy estimator to achieve optimal query complexity.
Polynomial Approximation Bounds for Monomials and Functions

This document details supporting material for research on polynomial approximation, particularly in the context of quantum algorithms and information theory. It establishes lower bounds on the degree required to approximate certain functions, such as monomials, and provides proofs for these bounds. The work explores how well even and odd functions can be approximated by polynomials with matching symmetry. The research demonstrates that even and odd functions can be accurately approximated by even and odd polynomials, respectively, without sacrificing approximation quality. This preservation of parity simplifies the approximation process and improves efficiency. The team also establishes a fundamental limit on how well monomials can be approximated by polynomials, proving that the degree of the approximating polynomial must grow at least proportionally to the square root of the monomial’s exponent. These results have implications for quantum algorithms, where polynomial approximation is used to represent complex functions in quantum circuits, and contribute to the broader field of approximation theory.
Tsallis Entropy Estimation Improves Approximation Bounds

Researchers have established a novel connection between information theory and approximation theory through new algorithms for estimating entropy. Their work provides an information-theoretic lower bound on the approximate degree of a monomial, surpassing previous bounds derived using Fourier analysis and established inequalities. This breakthrough stems from relating polynomial approximation of monomials to the estimation of Tsallis entropy, a measure of uncertainty in a system. The team developed an algorithm that estimates Tsallis entropy, for an unknown probability distribution or quantum state, to within a certain degree of error, using fewer computational queries than previously possible.

This represents a significant improvement over the Shift test, and, remarkably, is the first entropy estimator to achieve optimal query complexity for all relevant parameters simultaneously. Experiments demonstrate that this algorithm significantly reduces the number of measurements needed to accurately determine entropy, particularly for larger values of the entropy parameter, representing a substantial speedup. Furthermore, the researchers proved that their algorithm is optimal, differing from the theoretical lower bound by only a small, manageable factor, with implications for computing nonlinear functionals of quantum states and performing entanglement spectroscopy.
Optimal Entropy Estimation via Approximation Theory



This research establishes a new connection between information theory and approximation theory, specifically through algorithms designed to estimate entropy. The team demonstrates an information-theoretic lower bound for approximating the degree of a monomial, a result achieved by linking polynomial approximation to the estimation of Tsallis entropy. This work yields an algorithm that estimates Tsallis entropy, a generalization of the more common Shannon entropy, to within a certain degree of error, using fewer computational queries than previously possible, and represents the first entropy estimator with optimal query complexity across all relevant parameters. The findings improve upon existing methods for estimating Tsallis entropy, achieving greater efficiency in certain ranges of the entropy parameter. While the research provides tight bounds for a wide range of parameter values, the authors acknowledge that estimating Tsallis entropy for higher values remains an open problem, suggesting future research directions include investigating the computational complexity of estimating Tsallis entropy for these higher values and exploring the computational hardness of estimating Rényi entropy.

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