Quantum algorithm to distinguish Boolean functions of different weights

Abstract

We exploit Grover operator of database search algorithm for weight decision algorithm. In this research, weight decision problem is to find an exact weight w from given two weights as w1 and w2 where w1+w2=1 and 0<w1<w2<1. Firstly, if a Boolean function is given and when weights are {1/4,3/4}, we can find w with only one application of Grover operator. Secondly, if we apply k many times of Grover operator, we can decide w from the set of weights {sin^2(\frac{k}{2k+1}\frac{\pi}{2}) cos^2(\frac{k}{2k+1}\frac{\pi}{2})}. Finally, by changing the last two Grover operators with two phase conditions, we can decide w from given any set of two weights. To decide w with a sure success, if the quantum algorithm requires O(k) Grover steps, then the best known classical algorithm requires \Omega(k^s) steps where s>2. Hence the quantum algorithm achieves at least quadratic speedup.