A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions

TL;DR

This paper refines the understanding of the minimal degree of epsilon-error polynomials approximating symmetric Boolean functions by leveraging the connection between polynomials and quantum algorithms, removing logarithmic factors from previous bounds.

Contribution

It provides a tighter bound on polynomial degrees for symmetric functions by utilizing quantum algorithm techniques, improving upon Sherstov's recent characterization.

Findings

Tighter bounds on polynomial degrees without log-factors.

Utilization of quantum algorithms to analyze polynomial approximation.

Enhanced understanding of symmetric function approximation complexity.

Abstract

The degrees of polynomials representing or approximating Boolean functions are a prominent tool in various branches of complexity theory. Sherstov recently characterized the minimal degree deg_{\eps}(f) among all polynomials (over the reals) that approximate a symmetric function f:{0,1}^n-->{0,1} up to worst-case error \eps: deg_{\eps}(f) = ~\Theta(deg_{1/3}(f) + \sqrt{n\log(1/\eps)}). In this note we show how a tighter version (without the log-factors hidden in the ~\Theta-notation), can be derived quite easily using the close connection between polynomials and quantum algorithms.