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

The degrees of polynomials representing or approximating Boolean functions are aprominent tool in various branches of complexity theory. Sherstov [31] recently characterizedthe minimal degree dege(f) among all polynomials (over R) that approximatea symmetric function f : {0, 1}n → {0, 1} up to worst-case error e: dege(f) =Θ(deg1/3(f) + √n log(1/e). In this note we show how a tighter version (without thelog-factors hidden in the Θ-notation), can be derived quite easily using the close connectionbetween polynomials and quantum algorithms.

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