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. 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).