Quantum algorithm for the asymmetric weight decision problem and its generalization to multiple weights

As one of the applications of Grover search, an exact quantum algorithm for the symmetric weight decision problem of a Boolean function has been proposed recently. Although the proposed method shows a quadratic speedup over the classical approach, it only applies to the symmetric case of a Boolean function whose weight is one of the pair {0 < w1 < w2 < 1, w1 + w2 = 1}. In this article, we generalize this algorithm in two ways. Firstly, we propose a quantum algorithm for the more general asymmetric case where {0 < w1 < w2 < 1}. This algorithm is exact and computationally optimal. Secondly, we build on this to exactly solve the multiple weight decision problem for a Boolean function whose weight as one of {0 < w1 < w2 < … < wm < 1}. This extended algorithm continues to show a quantum advantage over classical methods. Thirdly, we compare the proposed algorithm with the quantum counting method. For the case with two weights, the proposed algorithm shows slightly lower complexity. For the multiple weight case, the two approaches show different performance depending on the number of weights and the number of solutions. For smaller number of weights and larger number of solutions, the weight decision algorithm can show better performance than the quantum counting method. Finally, we discuss the relationship between the weight decision problem and the quantum state discrimination problem.