Finite Domain Constraint Satisfaction Using Quantum Computation

We present a quantum algorithm for finite domain constraint solving, where the constraints have arity 2. It is complete and runs in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ O\left( {\left( {\left[ {d/2} \right]} \right)^{n/2} } \right) $$\end{document} time, where d is size of the domain of the variables and n the number of variables. For the case of d = 3 we provide a method to obtain an upper time bound of O(8n/8) ≈ O(1.2968n). Also for d = 5 the upper bound has been improved. Using this method in a slightly different way we can decide 3-colourability in O(1.2185n) time.