A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions
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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 $\mathbb{R}$) that approximate a symmetric function $f:\01^n\rightarrow\01$ up to worst-case error $\eps$: $ deg_{\eps}(f)=\widetilde{\Theta}\left(deg_{1/3}(f) + \sqrt{n\log(1/\eps)}\right).$ In this note we show how a tighter version (without the log-factors hidden in the $\widetilde{\Theta}$-notation), can be derived quite easily using the close connection between polynomials and quantum algorithms.
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2015 ◽
Vol DMTCS Proceedings, 27th...
(Proceedings)
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1969 ◽
Vol 12
(5)
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pp. 615-623
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1999 ◽
Vol 41
(1)
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pp. 125-128
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2012 ◽
Vol 22
(03)
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pp. 1250022
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2013 ◽
Vol 23
(02)
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pp. 386-398
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1978 ◽
Vol 84
(1)
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pp. 1-3
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