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Solving Hamiltonian Cycle Problem using Quantum $\mathbb{Z}_2$ Lattice Gauge Theory

The Hamiltonian cycle (HC) problem in graph theory is a well-known NP-complete problem. We present an approach in terms of $\mathbb{Z}_2$ lattice gauge theory (LGT) defined on the lattice with the graph as its dual. 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 spins in a same single-spin state, which can be obtained by using an adiabatic quantum algorithm with time complexity $O(\frac{1}{g_c^2} \sqrt{ \frac{1}{\varepsilon} N_e^{3/2}(N_v^3 + \frac{N_e}{g_c}}))$, where $N_v$ and $N_e$ are the numbers of vertices and edges of the graph respectively. A subsequent search for a HC among those closed-strings solves the HC problem. For some random samples of small graphs, we demonstrate that the dependence of the average value of $g_c$ on $\sqrt{N_{hc}}$, $N_{hc}$ being the number of HCs, and that of the average value of $\frac{1}{g_c}$ on $N_e$ are both linear. It is thus suggested that for some graphs, the HC problem may be solved in polynomial time. A possible quantum algorithm using $g_c$ to infer $N_{hc}$ is also discussed.