We show a connection between the deMorgan formula size of a Boolean function and the noise stability of the function. Using this connection, we show that the Fourier spectrum of any balanced Boolean function computed by a deMorgan formula of size s is concentrated on coefficients of degree up to O( √ s). These results have several applications that apply to any function f that can be computed by a deMorgan formula of size s. First, we get that f can be approximated (in L2-norm) with constant error by a polynomial of degree O( √ s). Second, we show an upper bound of O( √ s) on the average sensitivity of f . Our main result stems from a generalization of Khrapchenko’s bound [Khr71], that might be of independent interest, and some Fourier analysis on the Boolean cube. Previous works prove that any function f : {0, 1}n → {0, 1} that can be computed by a deMorgan formula of size s, can be approximated point-wise by a polynomial of degree O(s/2+o(1)) with constant point-wise error. We note that this result can be easily extended to have a polynomial of degree O(t ·s/2+o(1)) that approximates f with point-wise error 2−t, for any t > 0. This was shown in a long line of results in quantum complexity, including [BBC+01] and [FGG08, ACR+07, RS08, Rei09].
We show a connection between the deMorgan formula size of a Boolean function and the noise stability of the function. Using this connection, we show that the Fourier spectrum of any balanced Boolean function computed by a deMorgan formula of size s is concentrated on coefficients of degree up to O( √ s). These results have several applications that apply to any function f that can be computed by a deMorgan formula of size s. First, we get that f can be approximated (in L2-norm) with constant error by a polynomial of degree O( √ s). Second, we show an upper bound of O( √ s) on the average sensitivity of f . Our main result stems from a generalization of Khrapchenko’s bound [Khr71], that might be of independent interest, and some Fourier analysis on the Boolean cube. Previous works prove that any function f : {0, 1}n → {0, 1} that can be computed by a deMorgan formula of size s, can be approximated point-wise by a polynomial of degree O(s/2+o(1)) with constant point-wise error. We note that this result can be easily extended to have a polynomial of degree O(t ·s/2+o(1)) that approximates f with point-wise error 2−t, for any t > 0. This was shown in a long line of results in quantum complexity, including [BBC+01] and [FGG08, ACR+07, RS08, Rei09].
We prove that, to compute a Boolean function f on N variables with error probability ε, any quantum black-box algorithm has to query at least 21−2ερfN=21−2εSˉf times, where ρf is the average influence of variables in f, and Sˉf is the average sensitivity. It's interesting to contrast this result with the known lower bound of Ω(Sf), where Sf is the sensitivity of f. This lower bound is tight for some functions. We also show for any polynomial f~ that approximates f with error probability ε, deg(f~)≥1/4(1−1+ε3ε)2ρfN. This bound can be better than previous known lower bound of Ω(BSf) for some functions. Our technique may be of intest itself: we apply Fourier analysis to functions mapping {0,1}N to unit vectors in a Hilbert space. From this viewpoint, the state of the quantum computer at step t can be written as ∑s∈{0,1}N,∣s∣≤tϕ^s(−1)s⋅x, which is handy for lower bound analysis.
The approximate degree of a Boolean function f:{0,1}n→{0,1} is the minimum degree of a real polynomial p that approximates f pointwise: ∣f(x)−p(x)∣≤1/3 for all x∈{0,1}n. For every δ>0, we construct CNF and DNF formulas of polynomial size with approximate degree Ω(n1−δ), essentially matching the trivial upper bound of n. This improves polynomially on previous lower bounds and fully resolves the approximate degree of constant-depth circuits (AC0), a question that has seen extensive research over the past 10 years. Previously, an Ω(n1−δ) lower bound was known only for AC0 circuits of depth that grows with 1/δ (Bun and Thaler, FOCS 2017). Moreover, our CNF and DNF formulas are the simplest possible in that they have constant width. Our result holds even for one-sided approximation, and has the following further consequences. (i) We essentially settle the communication complexity of AC0 circuits in the bounded-error quantum model, k-party number-on-the-forehead randomized model, and k-party number-on-the-forehead nondeterministic model: we prove that for every δ>0, these models require Ω(n1−δ), Ω(n/4kk2)1−δ, and Ω(n/4kk2)1−δ, respectively, bits of communication even for polynomial-size constant-width CNF formulas. (ii) In particular, we show that the multiparty communication class coNPk can be separated essentially optimally from NPk and BPPk by a particularly simple function, a polynomial-size constant-width CNF. (iii) We give an essentially tight separation, of O(1) versus Ω(n1−δ), for the one-sided versus two-sided approximate degree of a function; and O(1) versus Ω(n1−δ) for the one-sided approximate degree of a function f versus its negation ¬f.
Exact and point-wise approximating representations of Boolean functions by real polynomials have been of great interest in the theory of computing. We focus on the study of sparsity of such representations. Our results include the following: First, we show that for every total Boolean function, its exact and approximate sparsity in the De Morgan basis are polynomially related to each other in the log scale, ignoring poly-log(n) factors. This answers an open question posed by Knop, Lovett, McGuire and Yuan (STOC 2021). It builds on and is analogous to the seminal result of Nisan and Szegedy (Computational Complexity 1994) who proved the same for degree and approximate degree. Second, we consider more powerful representations using generalized monomials, where each monomial is an indicator of a sub-cube. There are 3n such monomials, where n is the number of variables. We prove that even for these representations, the sparsity and approximate sparsity of total Boolean functions remain polynomially related to each other in the log scale, ignoring poly-log(n) factors. Third, we show that for every total Boolean function f, the log of its De Morgan sparsity characterizes up to polynomial loss and ignoring poly-log(n) factors, the quantum and classical 2-party bounded-error communication complexity of f ∘ EQ4, where EQ4 is Equality of two 2-bit strings, one held by Alice and the other by Bob. As a consequence, we show that bounded-error quantum protocols cannot exhibit super-polynomial cost advantage over their classical counterparts, for computing such functions.
The approximate degree of a Boolean function f(x1,x2,…,xn) is the minimum degree of a real polynomial that approximates f pointwise within 1/3. Upper bounds on approximate degree have a variety of applications in learning theory, differential privacy, and algorithm design in general. Nearly all known upper bounds on approximate degree arise in an existential manner from bounds on quantum query complexity. We develop a first-principles, classical approach to the polynomial approximation of Boolean functions. We use it to give the first constructive upper bounds on the approximate degree of several fundamental problems: - O(n43−4(2k−1)1) for the k-element distinctness problem; - O(n1−k+11) for the k-subset sum problem; - O(n1−k+11) for any k-DNF or k-CNF formula; - O(n3/4) for the surjectivity problem. In all cases, we obtain explicit, closed-form approximating polynomials that are unrelated to the quantum arguments from previous work. Our first three results match the bounds from quantum query complexity. Our fourth result improves polynomially on the Θ(n) quantum query complexity of the problem and refutes the conjecture by several experts that surjectivity has approximate degree Ω(n). In particular, we exhibit the first natural problem with a polynomial gap between approximate degree and quantum query complexity.
We describe a new hardness amplification result for point-wise approximation of Boolean functions by low-degree polynomials. Specifically, for any function f on N bits, define F(x_1,...,x_M) = OMB(f(x_1),...,f(x_M)) to be the function on M*N bits obtained by block-composing f with a function known as ODD-MAX-BIT. We show that, if f requires large degree to approximate to error 2/3 in a certain one-sided sense (captured by a complexity measure known as positive one-sided approximate degree), then F requires large degree to approximate even to error 1-2^{-M}. This generalizes a result of Beigel (Computational Complexity, 1994), who proved an identical result for the special case f=OR. Unlike related prior work, our result implies strong approximate degree lower bounds even for many functions F that have low threshold degree. Our proof is constructive: we exhibit a solution to the dual of an appropriate linear program capturing the approximate degree of any function. We describe several applications, including improved separations between the complexity classes P^{NP} and PP in both the query and communication complexity settings. Our separations improve on work of Beigel (1994) and Buhrman, Vereshchagin, and de Wolf (CCC, 2007).
Recent breakthroughs in quantum query complexity have shown that any formula of size n can be evaluated with O(sqrt(n)log(n)/log log(n)) many quantum queries in the bounded-error setting [FGG08, ACRSZ07, RS08b, Rei09]. In particular, this gives an upper bound on the approximate polynomial degree of formulas of the same magnitude, as approximate polynomial degree is a lower bound on quantum query complexity [BBCMW01]. These results essentially answer in the affirmative a conjecture of O'Donnell and Servedio [O'DS03] that the sign degree--the minimal degree of a polynomial that agrees in sign with a function on the Boolean cube--of every formula of size n is O(sqrt(n)). In this note, we show that sign degree is super-multiplicative under function composition. Combining this result with the above mentioned upper bounds on the quantum query complexity of formulas allows the removal of logarithmic factors to show that the sign degree of every size n formula is at most sqrt(n).