Ising Models for Solving the N-Queens Puzzle Based on the Domain-Wall Vectors

An Ising model is a mathematical model defined by an objective function comprising a quadratic formula of multiple spin variables, each taking values of either $-1$ or $+1$. The task of determining a spin value assignment to these variables that minimizes the resulting value of an Ising model is a challenging optimization problem. Recently, quantum annealers, consisting of qubit cells interconnected according to principles of quantum mechanics, have emerged as a solution for tackling such problems. Ising models characterized by fewer quadratic terms are preferable as they reduce the resource requirements of quantum annealers. Additionally, it is advantageous for the absolute values of coefficients associated with linear and quadratic terms to be small to facilitate the discovery of good solutions, given the inherent limitations in the resolution of quantum annealers. The primary contribution of this article lies in presenting Ising models tailored for solving the $n$-Queens puzzle. The conventional Ising model for this puzzle involves $\frac{5}{3}{n}^3-2n^2+\frac{n}{3}$ quadratic terms, with the maximum absolute value of coefficients being $4n+(nmod2)-7$. Our novel Ising model significantly reduces the number of quadratic terms to only $12n^2-24n+12$, with a maximum absolute coefficient of 6. Furthermore, we provide embedding results for a quantum annealer D-Wave Advantage utilizing a Pegasus graph $P\left(16\right)$. We succeeded in embedding our novel Ising model for up to the 21-Queens puzzle, while the conventional Ising model can be embedded only for up to the 14-Queens puzzle.