We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottomlevel 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 worstcase 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 linearsize 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 subexponentialtime algorithm was not known even for linearsize depth3 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).
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The Polynomial Method Strikes Back: Tight Quantum Query Bounds via Dual Polynomials
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.
kDistinctness: For any constant k, the approximate degree and quantum query complexity of the kdistinctness 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 kdistinctness 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.
kJunta testing: A tight Ω~(k1/2) lower bound for kjunta 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 lowdegree polynomials. Our lower bounds are proved by significantly refining techniques recently introduced by Bun and Thaler (FOCS 2017).
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 Award ID(s):
 1947889
 NSFPAR ID:
 10205756
 Date Published:
 Journal Name:
 Theory of computing
 Volume:
 16
 Issue:
 2020
 ISSN:
 15572862
 Page Range / eLocation ID:
 172
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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