Exact quantum algorithm to distinguish Boolean functions of different weights

Abstract In this work, we exploit the Grover operator for the weight analysis of a Boolean function, specifically to solve the weight-decision problem. The weight w is the fraction of all possible inputs for which the output is 1. The goal of the weight-decision problem is to find the exact weight w from the given two weights w1 and w2 satisfying a general weight condition as w1 + w2 = 1 and 0 < w1 < w2 < 1. First, we propose a limited weightdecision algorithm where the function has another constraint: a weight is in { w1 = sin2 ( k 2k+1 π 2 ) , w2 = cos2 ( k 2k+1 π 2 )} for integer k. Second, by changing the phases in the last two Grover iterations, we propose a general weightdecision algorithm which is free from the above constraint. Finally, we show that when our algorithm requires O(k) queries to find w with a unit success probability, any classical algorithm requires at least (k2) queries for a unit success probability. In addition, we show that our algorithm requires fewer queries to solve this problem compared with the quantum counting algorithm.