Hybrid Phase-Based Representation of Quantum Images

We propose a novel and hybrid quantum-classical algorithm that requires only O(logHW)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\log {HW})$$\end{document} qubits and reduces the multi-qubit gate costs required to represent an image of dimension (H×W)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(H \times W)$$\end{document}. In this algorithm, no qubit is needed to store the color information of the image. We represent the location information of an image with a superposition of mutually orthogonal vectors of an arbitrary basis and store the pixel information in the phases of the corresponding basis vectors without any extra qubit cost. We further present a classical algorithm to encode the phases and show that the inclusion of the classical algorithm significantly reduces the number of multi-qubit quantum gates required for image representation. Finally, we implement our algorithm on the classical simulator provided by IBM quantum as a proof of concept.