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 $\Theta(k)$ quantum property tester for this problem. With Gopalan \etal's classical lower bound of $\Omega(2^{k/2})$ for this task, this demonstrates an exponential quantum advantage for this task. We complement this result with a $\tilde{O}(2^{k/2}/\epsilon)$ classical tester, improving the best known upper bound and thus showing the prior lower bound is essentially tight.

Quantum Physics (quant-ph); Computational Complexity (cs.CC); Data Structures and Algorithms (cs.DS)

arXiv:2609.25816 [quant-ph]

(or arXiv:2609.25816v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.25816 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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