This saddle point is the point of minimized energy magnitude, formula_1, for the ions in the potential field. The Paul trap is often described as a harmonic potential well that traps ions in two dimensions (assume formula_2 and formula_3 without loss of generality) and does not trap ions in the formula_4 direction. When multiple ions are at the saddle point and the system is at equilibrium, the ions are only free to move in formula_4. Therefore, the ions will repel each other and create a vertical configuration in formula_4, the simplest case being a linear strand of only a few ions. Coulomb interactions of increasing complexity will create a more intricate ion configuration if many ions are initialized in the same trap. Furthermore, the additional vibrations of the added ions greatly complicate the quantum system, which makes initialization and computation more difficult.