Quantum Algorithm for Distribution-Free Junta Testing

Abstract

Inspired by a recent classical distribution-free junta tester by Chen, Liu, Serverdio, Sheng, and Xie (STOC'18), we construct a quantum tester for the same problem with complexity O(k/ε), which constitutes a quadratic improvement.We also prove that there is no efficient quantum algorithm for this problem using quantum examples as opposed to quantum membership queries.This result was obtained independently from the O(k/ε) algorithm for this problem by

Distribution-Free Testing of Linear Functions on R^n

“…Distribution-free testing (for graph properties) was first defined by Goldreich et al [22], though the first distribution-free testers for non-trivial properties appeared much later in the work of Halevy and Kushilevitz [23]. Subsequently, distribution-free testers have been considered for a variety of Boolean functions including low-degree polynomials, dictators, and monotone functions [23], k-juntas [6,12,23,30], conjunctions, decision lists, and linear threshold functions [20], monotone and non-monotone monomials [16], and monotone conjunctions [14,20]. However, to our knowledge the only (partial) distribution-free tester for a class of function on the Euclidean space is due to Harms [24] who gave an efficient tester for half spaces, that is, functions f : R n → {0, 1} of the form f (x) = sgn(w ⊤ x − θ) for some w ∈ R n and θ ∈ R, over any rotationally invariant distribution.…”