On the Noise Stability of Small De Morgan Formulas

We show a connection between the De Morgan formula size of a Boolean function f : {0, 1}n → {0, 1} and the noise stability of the function. Specifically, we prove that for 0 < p ≤ 1/2 it holds that NSp(f) ≥ 1− 2p √ L(f) · ‖f‖2 · (1− ‖f‖2) where NSp(f) is the noise stability of f with noise parameter p, ‖f‖ is the L2 norm of f , and L(f) is the De Morgan formula size of f . This result stems from a generalization of Khrapchenko’s bound [Khr71], that might be of independent interest. Our main result implies the following lower bound: ∑ S⊆[n] δ|S|f̂(S)2 ≥ ‖f‖ − 1− δ 2 √ L(f)‖f‖2(1− ‖f‖2) for 0 ≤ δ ≤ 1, where f̂(S) is the Fourier coefficient of f at S. In particular, this bound implies a concentration result on the spectrum of Boolean functions that can be computed by small De Morgan formulas. Specifically, for any ε > 0, we show that ∑ S⊆[n], |S|<k f̂(S)2 ≥ ‖f‖2 (1− ε) where k is roughly 1 2ε √ L(f)1−‖f‖ 2 ‖f‖2 . We observe that this concentration result also stems from a relation between the average sensitivity of f and the original Khrapcheko bound. In addition, we show that the De Morgan formula size in the results mentioned above can be replaced by the square of the non-negative quantum adversary bound, thus giving a (possibly) tighter bound. ∗Weizmann Institute of Science, Rehovot 76100, Israel. Email: {anat.ganor, ilan.komargodski}@weizmann.ac.il. Research supported by an Israel Science Foundation grant and by the I-CORE Program of the Planning and Budgeting Committee and the Israel Science Foundation. †Centre for Quantum Technologies, National University of Singapore, Singapore 117543. Email: [email protected]. ‡Weizmann Institute of Science, Rehovot 76100, Israel and Institute for Advanced Study, Princeton, NJ. Email: [email protected]. Research supported by an Israel Science Foundation grant, by the ICORE Program of the Planning and Budgeting Committee and the Israel Science Foundation, and by NSF grants number CCF-0832797, DMS-0835373.