A quantum algorithm for counting zero-crossings

TL;DR

This paper introduces a quantum algorithm that efficiently counts zero-crossings in a sequence, generalizing the Bernstein-Vazirani problem, and provides a quantum circuit for Walsh-Hadamard transforms in sequency ordering, useful in signal processing.

Contribution

The paper presents a quantum algorithm requiring only one oracle query for zero-crossings counting and a quantum circuit for Walsh-Hadamard transforms in sequency ordering, advancing quantum signal processing methods.

Findings

Quantum algorithm solves zero-crossings counting with one oracle query.

Quantum circuit for Walsh-Hadamard transform in sequency ordering.

Potential applications in digital signal and image processing.

Abstract

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.