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arXiv:2109.06763 · Fundamenta Informaticae

Testing Boolean Functions Properties

Zhengwei Xie, Daowen Qiu, Guangya Cai, Jozef Gruska, Paulo Mateus

Published Nov 18, 2021• 0 citations
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Abstract

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 $\varepsilon$-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 $\varepsilon$-far from $g$, where $\varepsilon$ is the parameter. We present a one-sided error quantum algorithm to deal with this problem that has the query complexity $O(\frac{1}{\sqrt{\varepsilon}})$. Moreover, we show that our quantum algorithm is optimal. Afterwards we show that the classical randomized query complexity of this problem is $\Theta(\frac{1}{\varepsilon})$. 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 $|C(f,g)|=\varepsilon$ and $|C(f,g)|=1$ using six queries, while the classical deterministic query complexity for this problem is $\Theta(2^{n})$ queries. Finally, we propose a one-sided error quantum query algorithm for testing whether one Boolean function is balanced versus $\varepsilon$-far balanced using $O(\frac{1}{\varepsilon})$ 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 $O(1/\varepsilon^{2})$. 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

DOI
10.46298/.8500

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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\varepsilonε-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 fff, of nnn variables, is identical with a given function ggg or is ε\varepsilonε-far from ggg, where ε\varepsilonε is the parameter. We present a one-sided error quantum algorithm to deal with this problem that has the query complexity O(1ε)O(\frac{1}{\sqrt{\varepsilon}})O(ε​1​). Moreover, we show that our quantum algorithm is optimal. Afterwards we show that the classical randomized query complexity of this problem is Θ(1ε)\Theta(\frac{1}{\varepsilon})Θ(ε1​). Secondly, we consider the D-J problem from the perspective of functional correlations and let C(f,g)C(f,g)C(f,g) denote the correlation of fff and ggg. We propose an exact quantum algorithm for making distinction between ∣C(f,g)∣=ε|C(f,g)|=\varepsilon∣C(f,g)∣=ε and ∣C(f,g)∣=1|C(f,g)|=1∣C(f,g)∣=1 using six queries, while the classical deterministic query complexity for this problem is Θ(2n)\Theta(2^{n})Θ(2n) queries. Finally, we propose a one-sided error quantum query algorithm for testing whether one Boolean function is balanced versus ε\varepsilonε-far balanced using O(1ε)O(\frac{1}{\varepsilon})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)O(1/\varepsilon^{2})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\varepsilonε-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 fff, of nnn variables, is identical with a given function ggg or is ε\varepsilonε-far from ggg, where ε\varepsilonε is the parameter. We present a one-sided error quantum algorithm to deal with this problem that has the query complexity O(1ε)O(\frac{1}{\sqrt{\varepsilon}})O(ε​1​). Moreover, we show that our quantum algorithm is optimal. Afterwards we show that the classical randomized query complexity of this problem is Θ(1ε)\Theta(\frac{1}{\varepsilon})Θ(ε1​). Secondly, we consider the D-J problem from the perspective of functional correlations and let C(f,g)C(f,g)C(f,g) denote the correlation of fff and ggg. We propose an exact quantum algorithm for making distinction between ∣C(f,g)∣=ε|C(f,g)|=\varepsilon∣C(f,g)∣=ε and ∣C(f,g)∣=1|C(f,g)|=1∣C(f,g)∣=1 using six queries, while the classical deterministic query complexity for this problem is Θ(2n)\Theta(2^{n})Θ(2n) queries. Finally, we propose a one-sided error quantum query algorithm for testing whether one Boolean function is balanced versus ε\varepsilonε-far balanced using O(1ε)O(\frac{1}{\varepsilon})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)O(1/\varepsilon^{2})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.
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.
The query model (or black-box model) has attracted much attention from the communities of both classical and quantum computing. Usually, quantum advantages are revealed by presenting a quantum algorithm that has a better query complexity than its classical counterpart. For example, the well-known quantum algorithms including Deutsch-Jozsa algorithm, Simon algorithm and Grover algorithm all show a considerable advantage of quantum computing from the viewpoint of query complexity. Recently we have considered in (Phys. Rev. A. {\bf 101}, 02232 (2020)) the problem: what functions can be computed by an exact one-query quantum algorithm? This problem has been addressed for total Boolean functions but still open for partial Boolean functions. Thus, in this paper we continue to characterize the computational power of exact one-query quantum algorithms for partial Boolean functions by giving several necessary and sufficient conditions. By these conditions, we construct some new functions that can be computed exactly by one-query quantum algorithms but have essential difference from the already known ones. Note that before our work, the known functions that can be computed by exact one-query quantum algorithms are all symmetric functions, whereas the ones constructed in this papers are generally asymmetric.
Given a Boolean function fff provided as a black-box with nnn variables, this paper will propose a quantum algorithm for testing if a certain variable is junta or εεε-far from being junta. The proposed algorithm constructs another black-box using two copies of the provided black-box. The constructed black-box is used with the partial diffusion operator in an amplitude amplification technique to test whether the variable being tested is junta or not, using O(2n)\mathcal{O}(\sqrt{2^n})O(2n​) queries to the constructed black-box. The proposed algorithm considers a Boolean function of general form, contrary to relevant algorithms proposed by others.
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 εεε-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 εεε-far (0<ε<12)(0 < ε< \frac{1}{2})(0<ε<21​) from linear functions requires O(ε−23)O(ε^{-\frac{2}{3}})O(ε−32​) many calls [Hillery and Andersson, Physical Review A 84, 062329 (2011)]. We show that this can be improved to O(ε−12)O(ε^{-\frac{1}{2}})O(ε−21​) by using the Deutsch-Jozsa and the Grover Algorithms.