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.
Categories: quant-ph