The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ 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: - $O\bigl(n^{\frac{3}{4}-\frac{1}{4(2^{k}-1)}}\bigr)$ for the $k$-element distinctness problem; - $O(n^{1-\frac{1}{k+1}})$ for the $k$-subset sum problem; - $O(n^{1-\frac{1}{k+1}})$ for any $k$-DNF or $k$-CNF formula; - $O(n^{3/4})$ 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.
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.
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 novel, first-principles 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; and 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.
Abstract. The approximate degree of a Boolean function [Formula: see text] is the minimum degree of a real polynomial [Formula: see text] that approximates [Formula: see text] pointwise: [Formula: see text] for all [Formula: see text]. For every [Formula: see text] we construct conjunctive normal form (CNF) and disjunctive normal form (DNF) formulas of polynomial size with approximate degree [Formula: see text] essentially matching the trivial upper bound of [Formula: see text]. This improves polynomially on previous lower bounds and fully resolves the approximate degree of constant-depth circuits [Formula: see text] a question that has seen extensive research over the past 10 years. Prior to our work, an [Formula: see text] lower bound was known only for [Formula: see text] circuits of depth that grows with [Formula: see text] (M. Bun and J. Thaler, SIAM J. Comput., 49 (2020), pp. FOCS17-59–FOCS17-96). Furthermore, the CNF and DNF formulas that we construct are the simplest possible in that they have constant width. Our proof departs significantly from previous approaches and contributes a novel, number-theoretic method for amplifying approximate degree. Our main result remains valid even for one-sided approximation: for any [Formula: see text], we construct a polynomial-size constant-width CNF formula with one-sided approximate degree [Formula: see text]. We thus obtain the following nearly tight separations: [Formula: see text] versus [Formula: see text] for the one-sided versus two-sided approximate degree of a function and [Formula: see text] versus [Formula: see text] for the one-sided approximate degree of a function [Formula: see text] versus its negation [Formula: see text]. As an application, we essentially settle the communication complexity of [Formula: see text] circuits in the bounded-error quantum model, [Formula: see text]-party number-on-the-forehead randomized model, and [Formula: see text]-party number-on-the-forehead nondeterministic model: we prove that for every [Formula: see text], these models require [Formula: see text], [Formula: see text], and [Formula: see text], respectively, bits of communication even for polynomial-size constant-width CNF formulas. In particular, we show that the multiparty communication class [Formula: see text] can be separated near-optimally from [Formula: see text] and [Formula: see text] by a particularly simple function, a polynomial-size constant-width CNF formula.
The ability (or inability) to represent or approximate Boolean functions by polynomials is a central concept in complexity theory, underlying interactive and probabilistically checkable proof systems, circuit lower bounds, quantum complexity theory, and more. In this book, the authors survey what is known about a particularly natural notion of approximation by polynomials, capturing pointwise approximation over This book covers recent progress on proving approximate degree lower and upper bounds and describes some applications of the new bounds to oracle separations, quantum query and communication complexity, and circuit complexity. The authors explain how several of these advances have been unlocked by a particularly simple and elegant technique, called dual block composition, for constructing solutions to this dual linear program. They also provide concise coverage of even more recent lower bound techniques based on a new complexity measure called spectral sensitivity. Finally, they show how explicit constructions of approximating polynomials have been inspired by quantum query algorithms. This book provides a comprehensive review of the foundational and recent developments of an important topic in both classical and quantum computing. The reader has a considerable body of knowledge condensed in an accessible form to quickly understand the principles and further their own research.
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.
The approximate non-deterministic degree of a Boolean function f, denoted ndegε(f) (written Nε(f) for brevity), is the minimum degree of a real polynomial p such that 0≤∣p(x)∣≤ε whenever f(x)=0, and ∣p(x)∣≥1 whenever f(x)=1. Unlike exact non-deterministic degree, which only requires the polynomial to be nonzero on 1-inputs, this measure enforces a uniform gap: the polynomial must stay close to zero on all 0-inputs and bounded away from zero on all 1-inputs.
The rational degree conjecture, open for over three decades, was recently resolved by Kothari, Kovacs-Deak, Wang, and Yang, who showed that for every total Boolean function f, deg(f)≤O(rdeg(f)3).
In their paper, they explicitly propose a stronger conjecture: that approximate degree is polynomially bounded by Nε(f) and Nε(f) jointly, i.e., for every total Boolean function f and every constant 0<ε<1, deg(f)≤poly(Nε(f),Nε(f)).
This conjecture, if true, would imply a polynomial version of the rational degree result and bring us closer to resolving de Wolf's longstanding non-deterministic degree conjecture.
In this work, we make the first systematic progress on this problem, establishing the conjecture for several broad and natural function classes: monotone and unate functions, functions of bounded alternation number, symmetric functions, k-uniform hypergraph properties, and read-k Disjunctive Normal Form (DNF) formulas.