We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let f be an m -bit Boolean function and consider an n -bit function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q(f) , we give an algorithm for evaluating F using O~(Q(f)⋅n) quantum queries. This improves on the bound of O(Q(f)⋅n) that follows by treating each conjunction independently, and our bound is tight for worst-case choices of f . Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f .By recursively applying our composition theorems, we obtain a nearly optimal O~(n1−2−d) upper bound on the quantum query complexity and approximate degree of linear-size depth- d AC 0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC 0 circuits.As an additional consequence, we show that AC 0∘⊕ circuits of depth d+1 require size Ω~(n1/(1−2−d))≥ω(n1+2−d) to compute the Inner Product function even on average. The previous best size lower bound was Ω(n1+4−(d+1)) and only held in the worst case (Cheraghchi et al., JCSS 2018).
We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let f be an m -bit Boolean function and consider an n -bit function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q(f) , we give an algorithm for evaluating F using O~(Q(f)⋅n) quantum queries. This improves on the bound of O(Q(f)⋅n) that follows by treating each conjunction independently, and our bound is tight for worst-case choices of f . Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f .By recursively applying our composition theorems, we obtain a nearly optimal O~(n1−2−d) upper bound on the quantum query complexity and approximate degree of linear-size depth- d AC 0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC 0 circuits.As an additional consequence, we show that AC 0∘⊕ circuits of depth d+1 require size Ω~(n1/(1−2−d))≥ω(n1+2−d) to compute the Inner Product function even on average. The previous best size lower bound was Ω(n1+4−(d+1)) and only held in the worst case (Cheraghchi et al., JCSS 2018).
We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let f be a Boolean function and consider a function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q(f), we give an algorithm for evaluating F using quantum queries. This improves on the bound of that follows by treating each conjunction independently, and is tight for worst-case choices of f. Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f. By recursively applying our composition theorems, we obtain a nearly optimal Õ(n1–2–d) upper bound on the quantum query complexity and approximate degree of linear-size depth-d AC0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC0 circuits. We also show that any substantially faster learning algorithm will require fundamentally new techniques.
We prove lower bounds on complexity measures, such as the approximate degree of a Boolean function and the approximate rank of a Boolean matrix, using quantum arguments. We prove these lower bounds using a quantum query algorithm for the combinatorial group testing problem. We show that for any function f, the approximate degree of computing the OR of n copies of f is Omega(sqrt n) times the approximate degree of f, which is optimal. No such general result was known prior to our work, and even the lower bound for the OR of ANDs function was only resolved in 2013. We then prove an analogous result in communication complexity, showing that the logarithm of the approximate rank (or more precisely, the approximate gamma-2 norm) of F: X x Y to 0,1 grows by a factor of Omega (sqrtn) when we take the OR of n copies of F, which is also essentially optimal. As a corollary, we give a new proof of Razborov's celebrated Omega(sqrtn) lower bound on the quantum communication complexity of the disjointness problem. Finally, we generalize both these results from composition with the OR function to composition with arbitrary symmetric functions, yielding nearly optimal lower bounds in this setting as well.
Quantum query complexity enjoys a variety of pleasing joint computation properties: for example, a composition theorem asserting Q(f∘g)=Θ(Q(f)Q(g)) for all Boolean functions f and g; a direct sum theorem asserting that computing k copies of a function (or search problem) costs Ω(k) times as much as the cost of computing one copy; and a direct product theorem asserting that for Boolean functions, even succeeding at the direct sum problem with exponentially small probability still requires Ω(k) times the cost of computing one copy to bounded error.
However, all of these results are strictly for worst-case quantum query complexity. For example, if we have a fixed distribution μ over inputs, the direct sum theorem says nothing about the quantum query complexity of computing k copies of f when the input comes from the product distribution μk instead of being worst-case. (Note that while a standard Yao-type minimax theorem guarantees a hard distribution for the direct sum problem, there's no guarantee that this hard distribution is a product distribution.) A similar problem occurs for the direct product theorem and the composition theorem: none of these results respect distributions.
In this work, we give distributional joint computation lower bounds for the composition, direct sum, and direct product problems. Along the way, we introduce some new tools for handling quantum query lower bounds, including (a) a new ``multiplicative'' variant of the γ2 norm (which we use in place of the multiplicative adversary method for proving the direct product theorem), and (b) a new ``Shaltiel-free'' measure of quantum query complexity, which we show characterizes the composition behavior of distributional quantum query complexity and satisfies pleasing properties.
We prove lower bounds on complexity measures, such as the approximate degree of a Boolean function and the approximate rank of a Boolean matrix, using quantum arguments. We prove these lower bounds using a quantum query algorithm for the combinatorial group testing problem. We show that for any function f, the approximate degree of computing the OR of n copies of f is Omega(sqrt{n}) times the approximate degree of f, which is optimal. No such general result was known prior to our work, and even the lower bound for the OR of ANDs function was only resolved in 2013. We then prove an analogous result in communication complexity, showing that the logarithm of the approximate rank (or more precisely, the approximate gamma_2 norm) of F: X x Y -> {0,1} grows by a factor of Omega~(sqrt{n}) when we take the OR of n copies of F, which is also essentially optimal. As a corollary, we give a new proof of Razborov's celebrated Omega(sqrt{n}) lower bound on the quantum communication complexity of the disjointness problem. Finally, we generalize both these results from composition with the OR function to composition with arbitrary symmetric functions, yielding nearly optimal lower bounds in this setting as well.
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.