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

M. Bun and J. Thaler

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 = \Omega(n^{1/3} * d^{2/3}). In particular, if d= n^{1-\Omega(1)}, then D is polynomially larger than d. Moreover, if f is computed by a polynomial-size Boolean circuit of constant depth, then so is F.

By recursively applying our transformation, 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 (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 delta > 0, an \Omega(n^{1-\delta}) 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.

Versions:

• FOCS 2017. [ECCC] [slides] [BIRS Talk Video (45 minutes)]

• These lecture notes elucidate various aspects of the proof. Whereas the proof in the paper takes place entirely in the "dual" world, the notes explain how to replace a key step of the proof with a "primal" argument that many will find more intuitive.