关于公式符号度的一个注记
A note on the sign degree of formulas
摘要：
最近量子查询复杂性的突破表明，任何大小为n的公式都可以在有限错误设置中用O（sqrt（n）log（n）/log log（n））来计算许多量子查询[FGG 08，ACRSZ 07，RS 08 b，Rei 09]。特别地，这给出了相同大小的公式的近似多项式次数的上限，因为近似多项式次数是量子查询复杂度的下限[BBCMW 01]。这些结果基本上回答了肯定的猜想O 'Donnell和Servedio [O' DS 03]的符号度-最小程度的多项式同意在签署与功能的布尔立方体-每一个公式的大小为n是O（sqrt（n））。本文证明了符号度在函数复合下是超乘法的。将该结果与上述公式的量子查询复杂度的上界相结合，可以去除对数因子，以表明每个大小为n的公式的符号度至多为sqrt（n）。
英文摘要：
Recent breakthroughs in quantum query complexity have shown that any formula of size n can be evaluated with O(sqrt(n)log(n)/log log(n)) many quantum queries in the bounded-error setting [FGG08, ACRSZ07, RS08b, Rei09]. In particular, this gives an upper bound on the approximate polynomial degree of formulas of the same magnitude, as approximate polynomial degree is a lower bound on quantum query complexity [BBCMW01]. These results essentially answer in the affirmative a conjecture of O'Donnell and Servedio [O'DS03] that the sign degree--the minimal degree of a polynomial that agrees in sign with a function on the Boolean cube--of every formula of size n is O(sqrt(n)). In this note, we show that sign degree is super-multiplicative under function composition. Combining this result with the above mentioned upper bounds on the quantum query complexity of formulas allows the removal of logarithmic factors to show that the sign degree of every size n formula is at most sqrt(n).
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