Let a Boolean function be available as a black-box (oracle) and one likes to devise an algorithm to test whether it has certain property or it is $ε$-far from having that property. The efficiency of the algorithm is judged by the number of calls to the oracle so that one can decide, with high probability, between these two alternatives. The best known quantum algorithm for testing whether a function is linear or $ε$-far $(0 < ε< \frac{1}{2})$ from linear functions requires $O(ε^{-\frac{2}{3}})$ many calls [Hillery and Andersson, Physical Review A 84, 062329 (2011)]. We show that this can be improved to $O(ε^{-\frac{1}{2}})$ by using the Deutsch-Jozsa and the Grover Algorithms.
Let a Boolean function be available as a black-box (oracle) and one likes to devise an algorithm to test whether it has certain property or it is ε-far from having that property. The efficiency of the algorithm is judged by the number of calls to the oracle so that one can decide, with high probability, between these two alternatives. The best known quantum algorithm for testing whether a function is linear or ε-far (0<ε<21) from linear functions requires O(ε−32) many calls [Hillery and Andersson, Physical Review A 84, 062329 (2011)]. We show that this can be improved to O(ε−21) by using the Deutsch-Jozsa and the Grover Algorithms.
The goal in function property testing is to determine whether a black-box Boolean function has a certain property or is $\ensuremath{\epsilon}$-far from having that property. The performance of the algorithm is judged by how many calls need to be made to the black box in order to determine, with high probability, which of the two alternatives is the case. Here we present two quantum algorithms, the first to determine whether the function is linear and the second to determine whether it is symmetric (invariant under permutations of the arguments). Both require order ${\ensuremath{\epsilon}}^{\ensuremath{-}2/3}$ calls to the oracle, which is better than known classical algorithms. In addition, in the case of linearity testing, if the function is linear, the quantum algorithm identifies which linear function it is. The linearity test combines the Bernstein-Vazirani algorithm and amplitude amplification, while the test to determine whether a function is symmetric uses projective measurements and amplitude amplification.
Boolean functions are important building blocks in cryptography for their wide application in both stream and block cipher systems. For cryptanalysis of such systems one tries to find out linear functions that are correlated to the Boolean functions used in the crypto system. Let f be an n-variable Boolean function and its Walsh spectra is denoted by Wf(ω) at the point ω∈{0,1}n. The Boolean function is available in the form of an oracle. We like to find an ω such that Wf(ω)=0 as this will provide one of the linear functions which are correlated to f. We show that the quantum algorithm proposed by Deutsch and Jozsa (1992) solves the above mentioned problem in constant time. However, the best known classical algorithm to solve this problem requires exponential time in n. We also analyse certain classes of cryptographically significant Boolean functions and highlight how the basic Deutsch-Jozsa algorithm performs on them.
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 ε-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 ε-far from g, where ε is the parameter. We present a one-sided error quantum algorithm to deal with this problem that has the query complexity O(ε1). Moreover, we show that our quantum algorithm is optimal. Afterwards we show that the classical randomized query complexity of this problem is Θ(ε1). 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)∣=ε and ∣C(f,g)∣=1 using six queries, while the classical deterministic query complexity for this problem is Θ(2n) queries. Finally, we propose a one-sided error quantum query algorithm for testing whether one Boolean function is balanced versus ε-far balanced using O(ε1) 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/ε2). Also this randomized algorithm is optimal. Besides, we link the problems considered here together and generalize them to the general case. Comment: 24 pages, comments are welcome. It is accepted for publishing in Fundamenta Informaticae
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 ε-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 ε-far from g, where ε is the parameter. We present a one-sided error quantum algorithm to deal with this problem that has the query complexity O(ε1). Moreover, we show that our quantum algorithm is optimal. Afterwards we show that the classical randomized query complexity of this problem is Θ(ε1). 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)∣=ε and ∣C(f,g)∣=1 using six queries, while the classical deterministic query complexity for this problem is Θ(2n) queries. Finally, we propose a one-sided error quantum query algorithm for testing whether one Boolean function is balanced versus ε-far balanced using O(ε1) 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/ε2). Also this randomized algorithm is optimal. Besides, we link the problems considered here together and generalize them to the general case. Comment: 24 pages, comments are welcome. It is accepted for publishing in Fundamenta Informaticae
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.