Classification of data with a qudit, a geometric approach

Abstract

We propose a model for data classification using isolated quantum d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{d}$$\\end{document}-level systems or else qudits. The procedure consists of an encoding phase where classical data are mapped on the surface of the qudit’s Bloch hyper-sphere via rotation encoding, followed by a rotation of the sphere and a projective measurement. The rotation is adjustable in order to control the operator to be measured, while additional weights are introduced in the encoding phase adjusting the mapping on the Bloch’s hyper-surface. During the training phase, a cost function based on the average expectation value of the observable is minimized using gradient descent thereby adjusting the weights. Using examples and performing a numerical estimation of lossless memory dimension, we demonstrate that this geometrically inspired qudit model for classification is able to solve nonlinear classification problems using a small number of parameters only and without requiring entangling operations.