Distribution-free junta testing

Quantum Algorithm for Distribution-Free Junta Testing

“…We have constructed a quantum algorithm with complexity O(k/ε), which gives a quadratic improvement in terms of k when compared to [10]. In the algorithm, we assume we only have classical access to D. It is also possible to have a quantum sampler from D, that is, an oracle of the form…”

“…Results. Recently, Liu et al [10] constructed a distribution-free randomised junta tester with complexity O(k 2 )/ε. Our main result is a distribution-free quantum junta tester with a (slightly better than) quadratic improvement in k. Theorem 1.…”

“…[10], Lemma 3.2. arXiv version). Assume f : {0, 1} n → {0, 1} is ε-far from any k-junta with respect to D, and S is a subset of [n] of size at most k. Then,Pr x∼D, T ⊆[n]\S f (x) = f (x T ) ≥ ε/2,where x is sampled from D, and T is a uniformly random subset of [n] \ S.…”

Almost Optimal Distribution-free Junta Testing

“…In the distribution-free property testing, [28], the distance between Boolean functions is measured with respect to an arbitrary and unknown distribution D over {0, 1} n . In this model, the testing algorithm is allowed (in addition to making black-box queries) to draw random x ∈ {0, 1} n according to the distribution D. This model is studied in [20,23,25,31,35]. For testing k-junta in this model, Chen et al [35] gave a onesided adaptive algorithm that makes Õ(k 2 )/ǫ queries and proved a lower bound Ω(2 k/3 ) for any non-adaptive algorithm.…”

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We consider the problem of testing whether an unknown n-variable Boolean function is a k-junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over {0, 1} n . Chen, Liu, Servedio, Sheng and Xie [35] showed that the distribution-free k-junta testing can be performed, with one-sided error, by an adaptive algorithm that makes Õ(k 2 )/ǫ queries. In this paper, we give a simple two-sided error adaptive algorithm that makes Õ(k/ǫ) queries.

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