The Quantum Adversary Method and Classical Formula Size Lower Bounds

We introduce two new complexity measures for Boolean functions, which we name sumPI and maxPI. The quantity sumPI has been emerging through a line of research on quantum query complexity lower bounds via the so-called quantum adversary method [Amb02, Amb03, BSS03, Zha05, LM04], culminating in [ˇSS05] with the realization that these many different formulations are in fact equivalent. Given that sumPI turns out to be such a robust invariant of a function, we begin to investigate this quantity in its own right and see that it also has applications to classical complexity theory. As a surprising application we show that sumPI2 (f) is a lower bound on the formula size, and even, up to a constant multiplicative factor, the probabilistic formula size of f. We show that several formula size lower bounds in the literature, specifically Khrapchenko and its extensions [Khr71, Kou93], including a key lemma of [H˚as98], are in fact special cases of our method. The second quantity we introduce,...

• Type: Conference
• Year: 2005
• Where: COCO
• Authors: Sophie Laplante, Troy Lee, Mario Szegedy
• Keywords: COCO 2005, Complexity Lower Bounds, Formula Size, Lower Bound