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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 ~ ( n 1 − 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 ~ ( n 1 / ( 1 − 2 − d ) ) ≥ ω ( n 1 + 2 − d ) to compute the Inner Product function even on average. The previous best size lower bound was Ω ( n 1 + 4 − ( d + 1 ) ) and only held in the worst case (Cheraghchi et al., JCSS 2018). 

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. k-Distinctness: For any constant k, the approximate degree and quantum query complexity of the k-distinctness 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 k-distinctness 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. k-Junta testing: A tight Ω~(k1/2) lower bound for k-junta testing, answering the main open question of Ambainis et al. (SODA 2016). Statistical distance from uniform: A tight Ω~(n1/2) lower bound for approximating the statistical distance of a distribution from uniform, answering the main question left open by Bravyi et al. (STACS 2010 and IEEE Trans. Inf. Theory 2011). Shannon entropy: A tight Ω~(n1/2) lower bound for approximating Shannon entropy up to a certain additive constant, answering a question of Li and Wu (2017). Surjectivity: The approximate degree of the surjectivity function is Ω~(n3/4). The best prior lower bound was Ω(n2/3). Our result matches an upper bound of O~(n3/4) due to Sherstov (STOC 2018), which we reprove using different techniques. The quantum query complexity of this function is known to be Θ(n) (Beame and Machmouchi, Quantum Inf. Comput. 2012 and Sherstov, FOCS 2015). Our upper bound for surjectivity introduces new techniques for approximating Boolean functions by low-degree polynomials. Our lower bounds are proved by significantly refining techniques recently introduced by Bun and Thaler (FOCS 2017). 

Recently, Bravyi, Gosset, and Konig (Science, 2018) exhibited a search problem called the 2D Hidden Linear Function (2D HLF) problem that can be solved exactly by a constant-depth quantum circuit using bounded fan-in gates (or QNC^0 circuits), but cannot be solved by any constant-depth classical circuit using bounded fan-in AND, OR, and NOT gates (or NC^0 circuits). In other words, they exhibited a search problem in QNC^0 that is not in NC^0. We strengthen their result by proving that the 2D HLF problem is not contained in AC^0, the class of classical, polynomial-size, constant-depth circuits over the gate set of unbounded fan-in AND and OR gates, and NOT gates. We also supplement this worst-case lower bound with an average-case result: There exists a simple distribution under which any AC^0 circuit (even of nearly exponential size) has exponentially small correlation with the 2D HLF problem. Our results are shown by constructing a new problem in QNC^0, which we call the Parity Halving Problem, which is easier to work with. We prove our AC^0 lower bounds for this problem, and then show that it reduces to the 2D HLF problem.