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

Shanghai Research Center for Quantum Science and CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China - Shanghai 201315, China, University of Science and Technology of China - Hefei 230026, China and Department of Physics & State Key Laboratory of Surface Physics, Fudan University - Shanghai 200433, China

Received: 20 October 2023
Accepted: 6 December 2023

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.