  We present a zero-crossings counting problem that is a generalization of the
Bernstein-Vazirani problem. The goal of this problem is to count the number of
zero-crossings (or sign changes) in a special type of sequence S, whose
definition depends upon a secret string. A quantum algorithm is presented to
solve this problem. The proposed quantum algorithm requires only one oracle
query to solve the problem, whereas a classical algorithm would need at least n
oracle queries, where $2^n$ is the size of the sequence S. In addition to
solving the zero-crossings counting problem, we also give a quantum circuit for
performing the Walsh-Hadamard transforms in sequency ordering. The
Walsh-Hadamard transform in sequency ordering is used in a wide range of
scientific and engineering applications, including in digital signal and image
processing. Therefore, the proposed quantum circuit for computing the
Walsh-Hadamard transforms in sequency ordering may be helpful in quantum
computing algorithms for applications for which the computation of the
Walsh-Hadamard transform in sequency ordering is required.
