Open Collections

Description

The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. For any constant $\delta > 0$, we exhibit an $AC^0$ function of approximate degree $\Omega(n^{1-delta})$. This improves over the best previous lower bound of $\Omega(n^{2/3})$ due to Aaronson and Shi, and nearly matches the trivial upper bound of n that holds for any function. We accomplish this by giving a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. I will also describe a number of applications of this result to communication complexity and cryptography. This is joint work with Mark Bun.