A note on the sign degree of formulas

TL;DR

This paper proves that the sign degree of formulas is at most proportional to the square root of their size, refining previous bounds and confirming a conjecture about their complexity.

Contribution

It establishes that sign degree is super-multiplicative under composition and removes logarithmic factors from bounds on formula sign degree.

Findings

Sign degree of formulas is at most sqrt(n).

Sign degree is super-multiplicative under composition.

Refines previous bounds on quantum query complexity and polynomial degree.

Abstract

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).