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.
The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically lifts to a quantum communication lower bound for a related function. We introduce a framework for proving approximate degree lower bounds for certain oracle identification problems, where the goal is to recover a hidden binary string
x∈{0,1}n given possibly non-standard oracle access to it. Our lower bounds apply to decision versions of these problems, where the goal is to compute the parity of
x. We apply our framework to the ordered search and hidden string problems, proving nearly tight approximate degree lower bounds of
Ω(n/log2n) for each. These lower bounds generalize to the weakly unbounded error setting, giving a new quantum query lower bound for the hidden string problem in this regime. Our lower bounds are driven by randomized communication upper bounds for the greater-than and equality functions.
The approximate degree of a Boolean function
f:{−1,1}n→{−1,1} is the least degree of a real polynomial that approximates
f pointwise to error at most
1/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function
f with approximate degree
d into a function
F on
O(n⋅polylog(n)) variables with approximate degree at least
D=Ω(n1/3⋅d2/3). In particular, if
d=n1−Ω(1), then
D is polynomially larger than
d. Moreover, if
f is computed by a polynomial-size Boolean circuit of constant depth, then so is
F. By recursively applying our transformation, for any constant
δ>0 we exhibit an AC
0 function of approximate degree
Ω(n1−δ). This improves over the best previous lower bound of
Ω(n2/3) due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of
n that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: * For any constant
δ>0, an
Ω(n1−δ) lower bound on the quantum communication complexity of a function in AC
0. * A Boolean function
f with approximate degree at least
C(f)2−o(1), where
C(f) is the certificate complexity of
f. This separation is optimal up to the
o(1) term in the exponent. * Improved secret sharing schemes with reconstruction procedures in AC
0.
The approximate degree of a Boolean function f : {-1, 1}n→ {-1, 1} is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function f with approximate degree d into a function F on O(n · polylog(n)) variables with approximate degree at least D = Ω(n1/3· d2/3). In particular, if d = n1-Ω(1), then D is polynomially larger than d. Moreover, if f is computed by a constant-depth polynomial-size Boolean circuit, then so is F. By recursively applying our transformation, for any constant δ > 0 we exhibit an AC0function of approximate degree Ω(n1-δ). This improves over the best previous lower bound of Ω(n2/3) due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of n that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: · For any constant δ > 0, an Ω(n1-δ) lower bound on the quantum communication complexity of a function in AC0. · A Boolean function f with approximate degree at least C(f)2-o(1), where C(f) is the certificate complexity of f. This separation is optimal up to the o(1) term in the exponent. · Improved secret sharing schemes with reconstruction procedures in AC0.