Quantum algorithms to check Resiliency, Symmetry and Linearity of a Boolean function

In this paper, we present related quantum algorithms to check the order of resiliency, symmetry and linearity of a Boolean function that is available as a black-box (oracle). First we consider resiliency and show that the Deutsch-Jozsa algorithm can be immediately used for this purpose. We also point out how the quadratic improvement in query complexity can be obtained over the DeutschJozsa algorithm for this purpose using the Grover’s technique. While the worst case quantum query complexity to check the resiliency order is exponential in the number of input variables of the Boolean function, we require polynomially many measurements only. We also describe a subset of n-variable Boolean functions for which the algorithm works in polynomially many steps, i.e., we can achieve an exponential speed-up over best known classical algorithms. A similar kind of approach can be exploited to check whether a Boolean function is symmetric (respectively linear) or not. Given a Boolean function as an oracle, it is important to devise certain algorithms to test whether it has a specific property or it is -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. We show that this can be achieved in O( − 1 2 ) query complexity. This is obtained by showing that the problem of checking symmetry or linearity can be efficiently reduced to testing whether a Boolean function is constant. This is a thoroughly revised and extended version of the paper “Quantum algorithms to check Resiliency of a Boolean function” by Kaushik Chakraborty and Subhamoy Maitra that has been presented in WCC 2013, April 15-19, 2013, Bergen, Norway. Sections 3, 4 contain additional materials over the workshop version. Kaushik Chakraborty Indian Statistical Institute, Kolkata 700 108, India E-mail: [email protected] Anupam Chattopadhyay UMIC Research Centre, RWTH Aachen University, Aachen, Germany E-mail: [email protected] Subhamoy Maitra Applied Statistics Unit, Indian Statistical Institute, Kolkata 700 108, India E-mail: [email protected] 2 Chakraborty, Chattopadhyay & Maitra