The Polynomial Method Strikes Back: Tight Quantum Query Bounds via Dual Polynomials

TL;DR

This paper advances the polynomial method to establish tight quantum query bounds for fundamental functions, resolving long-standing conjectures and introducing new techniques for polynomial approximation and lower bound proofs.

Contribution

It provides nearly tight bounds for approximate degree and quantum query complexity of key functions, and develops novel techniques for polynomial approximation and lower bounds.

Findings

Approximate degree of k-distinctness is (n^{3/4-1/(2k)}) for constant k.

Image size testing has (n^{1/2}) lower bound, confirming a conjecture.

Surjectivity's approximate degree is (n^{3/4}), matching known upper bounds.

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. Approximate degree is known to be a lower bound on quantum query complexity. We resolve or nearly resolve the approximate degree and quantum query complexities of the following basic functions: $\bullet$ $k$-distinctness: For any constant $k$, the approximate degree and quantum query complexity of $k$-distinctness is $\Omega(n^{3/4-1/(2k)})$. This is nearly tight for large $k$ (Belovs, FOCS 2012). $\bullet$ Image size testing: The approximate degree and quantum query complexity of testing the size of the image of a function $[n] \to [n]$ is $\tilde{\Omega}(n^{1/2})$. This proves a conjecture of Ambainis et al. (SODA 2016), and it implies the following lower bounds: $-$ $k$-junta testing: A tight $\tilde{\Omega}(k^{1/2})$ lower bound, answering the main open question of Ambainis et al. (SODA 2016). $-$ Statistical Distance from Uniform: A tight $\tilde{\Omega}(n^{1/2})$ lower bound, 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, answering a question of Li and Wu (2017). $\bullet$ 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, QIC 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).