Quantum algorithm for the asymmetric weight decision problem
and its generalization to multiple weights
Choi, B.-S.
and
Braunstein, S.L.

(2011):
*Quantum Information Processing*
**10**,
177-188
(PDF)
ABSTRACT:
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 < w*_{1} < w_{2} < 1,
*w*_{1} + w_{2} = 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 < w*_{1} < w_{2} < 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 < w*_{1} < w_{2} < …
< w_{m} < 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.