Correspondence between the Hamiltonian cycle problem and the quantum lattice gauge theory

Abstract

We propose the correspondence between the Hamiltonian cycle (HC) problem in graph theory and the quantum lattice gauge theory (QZ2LGT) defined on the lattice dual to that graph. For the QZ2LGT, when the coupling parameter g is less than the critical value g c , the ground state is a superposition of all configurations with closed strings of same spins, which can be obtained by using an adiabatic quantum algorithm. A subsequent search for a HC among those closed strings solves the original HC problem. The method is demonstrated for random samples of small graphs.