The goal in the area of functions property testing is to determine whether a given black-box Boolean function has a particular given property or is ɛ-far from having that property. We investigate here several types of properties testing for Boolean functions (identity, correlations and balancedness) using the Deutsch-Jozsa algorithm (for the Deutsch-Jozsa (D-J) problem) and also the amplitude amplification technique. At first, we study here a particular testing problem: namely whether a given Boolean function f, of n variables, is identical with a given function g or is ɛ-far from g, where ɛ is the parameter. We present a one-sided error quantum algorithm to deal with this problem that has the query complexity [Formula: see text]. Moreover, we show that our quantum algorithm is optimal. Afterwards we show that the classical randomized query complexity of this problem is [Formula: see text]. Secondly, we consider the D-J problem from the perspective of functional correlations and let C( f, g) denote the correlation of f and g. We propose an exact quantum algorithm for making distinction between | C( f, g)| = ɛ and | C( f, g)| = 1 using six queries, while the classical deterministic query complexity for this problem is Θ(2 n ) queries. Finally, we propose a one-sided error quantum query algorithm for testing whether one Boolean function is balanced versus ɛ-far balanced using [Formula: see text] queries. We also prove here that our quantum algorithm for balancedness testing is optimal. At the same time, for this balancedness testing problem we present a classical randomized algorithm with query complexity of O(1/ ɛ 2 ). Also this randomized algorithm is optimal. Besides, we link the problems considered here together and generalize them to the general case.

In thedistributed Deutsch–Jozsa promise problem, two parties are to determine whether their respective stringsx, y∈ {0,1}nare at theHamming distanceH(x, y) = 0 orH(x, y) = $\frac{n}{2}$ . Buhrmanet al.(STOC' 98) proved that the exactquantum communication complexityof this problem isO(logn) while thedeterministic communication complexityisΩ(n). This was the first impressive (exponential) gap between quantum and classical communication complexity. In this paper, we generalize the above distributed Deutsch–Jozsa promise problem to determine, for any fixed $\frac{n}{2}$ ⩽k⩽n, whetherH(x, y) = 0 orH(x, y) =k, and show that an exponential gap between exact quantum and deterministic communication complexity still holds ifkis an even such that $\frac{1}{2}$ n⩽k< (1 − λ)n, where 0 < λ < $\frac{1}{2}$ is given. We also deal with a promise version of the well-knowndisjointnessproblem and show also that for this promise problem there exists an exponential gap between quantum (and also probabilistic) communication complexity and deterministic communication complexity of the promise version of such a disjointness problem. Finally, some applications to quantum, probabilistic and deterministic finite automata of the results obtained are demonstrated.