A hybrid classical-quantum algorithm for digital image processing

Abstract

A hybrid classical-quantum approach for evaluation of multi-dimensional Walsh–Hadamard transforms and its applications to quantum image processing are proposed. In this approach, multi-dimensional Walsh–Hadamard transforms are obtained using quantum Hadamard gates (along with state preparation, shifting, scaling and measurement operations). The proposed approach for evaluation of multi-dimensional Walsh–Hadamard transform has a considerably lower computational complexity (involving O(Nd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(N^d)$$\end{document} operations) in contrast to classical Fast Walsh–Hadamard transform (involving O(Ndlog2Nd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(N^d~\log _2 N^d)$$\end{document} operations), where d and N denote the number of dimensions and degrees of freedom along each dimension. Unlike many other quantum image representation and quantum image processing frameworks, our proposed approach makes efficient use of qubits, where only log2N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log _2 N $$\end{document} qubits are sufficient for sequential processing of an image of N×N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ N \times N $$\end{document} pixels. Selected applications of the proposed approach (for d=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ d=2 $$\end{document}) are demonstrated via computational examples relevant to basic image filtering and periodic banding noise removal, and the results were found to be satisfactory.