Exponential Quantum Advantage in Testing Fourier Dimensionality

Abstract

A boolean function $f$ has Fourier dimension $k$ if its nonzero Fourier coefficients span a subspace of dimension $k$. We consider the property testing task of determining whether a function has Fourier dimension at most $k$, or is $\epsilon$-far from being so. We show that there is a $O(k/\sqrt{\epsilon})$-query quantum property tester for this problem, which we show to be almost optimal. Combined with Gopalan et al.'s classical lower bound of $\Omega(2^{k/2})$, this demonstrates an exponential quantum advantage for this task \cite{DBLP:journals/siamcomp/GopalanOSSW11}. We complement this result with a $\tilde{O}(2^{k/2}/\epsilon)$ classical tester, giving a quadratic improvement over the previous best tester, and essentially settling the classical query complexity.