caa a22 4500 467909725 CHVBK 20180406152846.0 cr unu---uuuuu 170328e20060601xx s 000 0 eng 10.1007/s00037-006-0212-7 doi (NATIONALLICENCE)springer-10.1007/s00037-006-0212-7 THE QUANTUM ADVERSARY METHOD AND CLASSICAL FORMULA SIZE LOWER BOUNDS [Elektronische Daten] [Sophie Laplante, Troy Lee, Mario Szegedy] Abstract.: 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 (Ambainis 2002, 2003; Barnum et al. 2003; Laplante & Magniez 2004; Zhang 2005), culminating in Špalek & Szegedy (2005) 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 sumPI 2(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 (Khrapchenko 1971; Koutsoupias 1993), including a key lemma of Håstad (1998), are in fact special cases of our method. The second quantity we introduce, maxPI (f), is always at least as large as sumPI(f) , and is derived from sumPI in such a way that maxPI 2(f) remains a lower bound on formula size. Our main result is proven via a combinatorial lemma which relates the square of the spectral norm of a matrix to the squares of the spectral norms of its submatrices. The generality of this lemma implies that our methods can also be used to lower-bound the communication complexity of relations, and a related combinatorial quantity, the rectangle partition number. To exhibit the strengths and weaknesses of our methods, we look at the sumPI and maxPI complexity of a few examples, including the recursive majority of three function, a function defined by Ambainis (2003), and the collision problem. Birkhäuser Verlag, Basel, 2006 Lower bounds nationallicence quantum computing nationallicence adversary method nationallicence formula size nationallicence communication complexity nationallicence 68Q17 nationallicence 68Q30 nationallicence Laplante Sophie LRI, Bâtiment 490, Université Paris-Sud, 91405, Orsay Cedex, France aut Lee Troy CWI and University of Amsterdam, 413 Kruislaan, 1098, Amsterdam, SJ, The Netherlands aut Szegedy Mario Department of Computer Science, Rutgers University, 110 Frelinghuysen Road, 08854-8019, Piscataway, NJ, U.S.A aut computational complexity Birkhäuser-Verlag; www.birkhauser.ch 15/2(2006-06-01), 163-196 1016-3328 15:2<163 2006 15 37 https://doi.org/10.1007/s00037-006-0212-7 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00037-006-0212-7 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Laplante Sophie LRI, Bâtiment 490, Université Paris-Sud, 91405, Orsay Cedex, France aut NATIONALLICENCE 700 1- Lee Troy CWI and University of Amsterdam, 413 Kruislaan, 1098, Amsterdam, SJ, The Netherlands aut NATIONALLICENCE 700 1- Szegedy Mario Department of Computer Science, Rutgers University, 110 Frelinghuysen Road, 08854-8019, Piscataway, NJ, U.S.A aut NATIONALLICENCE 773 0- computational complexity Birkhäuser-Verlag; www.birkhauser.ch 15/2(2006-06-01), 163-196 1016-3328 15:2<163 2006 15 37 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer