Graph comparison via nonlinear quantum search

Abstract

Graph comparison is an established NP-hard problem. In this paper, we present an efficiently scaling quantum algorithm which finds the size of the maximum common edge subgraph for any pair of unlabelled graphs and thus provides a meaningful measure of graph similarity. The algorithm makes use of a two-part quantum dynamic process: in the first part, we obtain information crucial for the comparison of two graphs through linear quantum computation. However, this information is hidden in the quantum system with such a vanishingly small amplitude that even quantum algorithms such as Grover’s search are not fast enough to distil it efficiently. In order to extract the information, we call upon techniques in nonlinear quantum computing to provide the speed-up necessary for an efficient algorithm. The linear quantum circuit requires O(n3log3(n)loglog(n))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(n^3 \log ^3 (n) \log \log (n))$$\end{document} elementary quantum gates, and the nonlinear evolution under the Gross–Pitaevskii equation has a time scaling of O(1gn2log3(n)loglog(n))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(\frac{1}{g} n^2 \log ^3 (n) \log \log (n))$$\end{document}, where n is the number of vertices in each graph and g is the strength of the Gross–Pitaevskii nonlinearity. Through this example, we demonstrate the power of nonlinear quantum search techniques to solve a subset of NP-hard problems.