A Nearly Optimal Lower Bound on the Approximate Degree of AC^0

Abstract

The approximate degree of a Boolean function f: {-1, 1}^n ↦ {-1, 1} is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits.Specifically, we show how to transform any Boolean function f with approximate degree d into a function F on O(n polylog(n)) variables with approximate degree at least D = Ω(n^{1/3} d^{2/3}). In particular, if d = n^{1-Ω(1), then D is polynomially larger than d. Moreover, if f is computed by a constant-depth polynomial-size Boolean circuit, then so is F.By recursively applying our transformation, for any constant δ 0 we exhibit an AC° function of approximate degree Ω(n^{1-δ}). This improves over the best previous lower bound of Ω(n^{2/3}) due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of n that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width.We describe several applications of these results. We give:• For any constant δ 0, an Ω(n^{1-δ}) lower bound on the quantum communication complexity of a function in AC°.• A Boolean function f with approximate degree at least C(f)^{2-o(1), where C(f) is the certificate complexity of f. This separation is optimal up to the o(1) term in the exponent.• Improved secret sharing schemes with reconstruction procedures in AC°.