The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. The approximate degree of f is known to be a lower bound on the quantum query complexity of f (Beals et al., FOCS 1998 and J. ACM 2001). We find tight or nearly tight bounds on the approximate degree and quantum query complexities of several basic functions. Specifically, we show the following. kDistinctness: For any constant k, the approximate degree and quantum query complexity of the kdistinctness function is Ω(n3/4−1/(2k)). This is nearly tight for large k, as Belovs (FOCS 2012) has shown that for any constant k, the approximate degree and quantum query complexity of kdistinctness is O(n3/4−1/(2k+2−4)). Image size testing: The approximate degree and quantum query complexity of testing the size of the image of a function [n]→[n] is Ω~(n1/2). This proves a conjecture of Ambainis et al. (SODA 2016), and it implies tight lower bounds on the approximate degree and quantum query complexity of the following natural problems. kJunta testing: A tight Ω~(k1/2) lower bound for kjunta testing, answering the main open question of Ambainis et al. (SODA 2016). Statistical distance frommore »
Approximating the Distance to Monotonicity of Boolean Functions
We design a nonadaptive algorithm that, given a Boolean function f: {0, 1}^n → {0, 1} which is αfar from monotone, makes poly(n, 1/α) queries and returns an estimate that, with high probability, is an Otilde(\sqrt{n})approximation to the distance of f to monotonicity. Furthermore, we show that for any constant k > 0, approximating the distance to monotonicity up to n^(1/2−k)factor requires 2^{n^k} nonadaptive queries, thereby ruling out a poly(n, 1/α)query nonadaptive algorithm for such approximations. This answers a question of Seshadhri (Property Testing Review, 2014) for the case of nonadaptive algorithms. Approximating the distance to a property is closely related to tolerantly testing that property. Our lower bound stands in contrast to standard (nontolerant) testing of monotonicity that can be done nonadaptively with Otilde(n/ε^2) queries.
We obtain our lower bound by proving an analogous bound for erasureresilient testers. An αerasureresilient tester for a desired property gets oracle access to a function that has at most an α fraction of values erased. The tester has to accept (with probability at least 2/3) if the erasures can be filled in to ensure that the resulting function has the property and to reject (with probability at least 2/3) if every completion of erasures more »
 Award ID(s):
 1909612
 Publication Date:
 NSFPAR ID:
 10185525
 Journal Name:
 Proceedings of the 2020 ACMSIAM Symposium on Discrete Algorithms, SODA 2020
 Page Range or eLocationID:
 19952009
 Sponsoring Org:
 National Science Foundation
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