Quantum algorithm for edge detection in digital grayscale images

TL;DR

This paper introduces a quantum algorithm for edge detection in grayscale images using a sequency-ordered Walsh-Hadamard transform, offering significant improvements in circuit depth and computational efficiency over existing quantum methods.

Contribution

The paper presents a novel quantum edge detection algorithm based on sequency-ordered Walsh-Hadamard transform with reduced circuit depth and computational cost compared to prior quantum techniques.

Findings

Achieves circuit depth of O(n), better than QFT's O(n^2)

Computational cost of O(log(N1N2)), outperforming QHED

Demonstrates effective edge detection through computational examples

Abstract

In this work, we propose a novel quantum algorithm for edge detection in digital grayscale images, based on the sequency-ordered Walsh-Hadamard transform. The proposed method significantly improves upon existing quantum techniques for edge detection by using a quantum algorithm for the sequency-ordered Walsh-Hadamard transform, achieving a circuit depth of $\mathcal{O}(n)$ (where $n$ is the number of qubits). This represents a notable enhancement over the Quantum Fourier Transform (QFT), which has a circuit depth of $\mathcal{O}(n^{2})$. Furthermore, our approach for edge detection has a computational cost (both gate complexity and quantum circuit depth) of $\mathcal{O}(\log_{2}(N_{1}N_{2}))$ for an image of size $N_{1}\times N_{2}$, offering a considerable improvement over the Quantum Hadamard Edge Detection (QHED) algorithm, which incurs a cost of $\mathcal{O}(\text{poly}(\log_{2}(N_{1}N_{2})))$. By integrating a quantum high-pass filter with the sequency-ordered Walsh-Hadamard transform, the algorithm effectively extracts edge information from images. Computational examples are provided to demonstrate the efficacy of the proposed algorithm which provides a better performance in comparison to QHED.