We construct efficient or query efficient quantum property testers for two existential group properties which have exponential query complexity both for their decision problem in the quantum and fo...
We construct efficient or query efficient quantum property testers for two existential group properties which have exponential query complexity both for their decision problem in the quantum and for their testing problem in the classical model of computing. These are periodicity in groups and the common coset range property of two functions having identical ranges within each coset of some normal subgroup. Our periodicity tester is efficient in Abelian groups and generalizes, in several aspects, previous periodicity testers. This is achieved by introducing a technique refining the majority correction process widely used for proving robustness of algebraic properties. The periodicity tester in non-Abelian groups and the common coset range tester are query efficient.
We construct efficient or query efficient quantum property testers for two existential group properties which have exponential query complexity both for their decision problem in the quantum and for their testing problem in the classical model of computing. These are periodicity in groups and the common coset range property of two functions having identical ranges within each coset of some normal subgroup. Our periodicity tester is efficient in Abelian groups and generalizes, in several aspects, previous periodicity testers. This is achieved by introducing a technique refining the majority correction process widely used for proving robustness of algebraic properties. The periodicity tester in non-Abelian groups and the common coset range tester are query efficient.
A boolean function f has Fourier dimension k if its strings with nonzero Fourier coefficients span a subspace of dimension k. We first consider the property testing task of determining whether a function has Fourier dimension at most k, or is ε-far from being so. We show that there is an O(k/ε)-query quantum property tester for this problem, which we show to be almost optimal. Combined with Gopalan \etal's classical lower bound of Ω(2k/2), this demonstrates an exponential quantum advantage for this task. We complement this result with a O~(2k/2/ε) classical tester, giving a quadratic improvement over the previous best tester, and essentially settling the classical query complexity.
We then introduce and study the tolerant version of this problem. That is, determining whether a function is αε-close to having Fourier dimension at most k, or is ε-far from being so. Perhaps surprisingly, we show that this tolerant version of the problem is essentially just as hard as the standard version of the problem. That is, we show this problem has a O~(k/ε) quantum tester, and a O~(2k/2/ε) classical tester.