The Polynomial Method Strikes Back:
Tight Quantum Query Bounds via Dual Polynomials
M. Bun, R. Kothari, and J. Thaler
Abstract:
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 resolve or nearly resolve 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 \Omega(n^{3/4-1/(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(n^{3/4-1/(2^{k+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 \tilde{\Omega}(n^{1/2}). 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 \tilde{\Omega}(k^{1/2}) lower bound for k-junta testing,
answering the main open question of Ambainis et al. (SODA 2016).
- Statistical Distance from Uniform: A tight \tilde{\Omega}(n^{1/2}) lower bound for
approximating the statistical distance from uniform of a distribution,
answering the main question left open by Bravyi et al. (STACS 2010 and IEEE Trans. Inf. Theory 2011).
- Shannon entropy: A tight \tilde{\Omega}(n^{1/2}) 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 \tilde{\Omega}(n^{3/4}).
The best prior lower bound was \Omega(n^{2/3}). Our result matches an upper bound of
\tilde{O}(n^{3/4}) due to Sherstov, which we reprove using different techniques.
The quantum query complexity of this function is known to be \Theta(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).
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