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

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
⁰
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
⁰
. · 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
⁰
.