Testing Boolean Functions Properties

Article type: Research Article

Authors: Zhengwei, Xie^a | Daowen, Qiu^{b; †; *} | Guangya, Cai^c | Gruska, Jozef^d | Mateus, Paulo^e

Affiliations: [a] School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou 510006, China. [email protected] | [b] School of Computer Science and Engineering, Sun Yat-sen Univ., Guangzhou 510006, China. [email protected] | [c] School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou 510006, China. [email protected] | [d] Faculty of Informatics, Masaryk University, Brno, Czech Republic. [email protected] | [e] Instituto de Telecomunicações, Dept. de Matematica, Instituto Superior Técnico, Av. Rovisco Pais 1049-001 Lisbon, Portugal. [email protected]

Correspondence: [*] Address for correspondence: School of Computer Sci. and Eng., Sun Yat-sen University, Guangzhou 510006, China.

Note: [†] Also works: Instituto de Telecomunicações, Dept. de Matematica, Instituto Superior Técnico, Lisbon, Portugal. This work is partly supported by the National Natural Science Foundation of China (Nos. 61572532, 61876195), the Natural Science Foundation of Guangdong Province of China (No. 2017B030311011).

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 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 O(1ε). Moreover, we show that our quantum algorithm is optimal. Afterwards we show that the classical randomized query complexity of this problem is Θ(1ε). 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(2n) queries. Finally, we propose a one-sided error quantum query algorithm for testing whether one Boolean function is balanced versus ɛ-far balanced using O(1ε) 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/ɛ2). Also this randomized algorithm is optimal. Besides, we link the problems considered here together and generalize them to the general case.

Keywords: Deutsch-Jozsa Algorithm, Quantum amplitude amplification, Identity testing, Correlation testing, Balancedness testing

DOI: 10.3233/FI-2021-2076

Journal: Fundamenta Informaticae, vol. 182, no. 4, pp. 321-344, 2021