Exact quantum algorithm to distinguish Boolean functions of different weights
Samuel L. Braunstein, Byung-Soo Choi, Subhroshekhar Ghosh, Subhamoy Maitra · Journal of Physics A Mathematical and Theoretical · 2007
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 w 1 and w 2 satisfying a general weight condition as w 1 + w 2 = 1 and 0 < w 1 < w 2 < 1. First, we propose a limited weight-decision algorithm where the function has another constraint: a weight is in for integer k . Second, by changing the phases in the last two Grover iterations, we propose a general weight-decision 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 Ω( k 2 ) 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.