In this paper, we explore the stability of the energy landscape of an Ising Hamiltonian when subjected to two kinds of perturbations: a perturbation on the coupling coefficients and external fields, and a perturbation on the underlying graph structure. We give sufficient conditions so that the ground states of a given Hamiltonian are stable under perturbations of the first kind in terms of order preservation. Here by order preservation we mean that the ordering of energy corresponding to two spin configurations in a perturbed Hamiltonian will be preserved in the original Hamiltonian up to a given error margin. We also estimate the probability that the energy gap between ground states for the original Hamiltonian and the perturbed Hamiltonian is bounded by a given error margin when the coupling coefficients and local external magnetic fields of the original Hamiltonian are i.i.d. Gaussian random variables. In the end we show a concrete example of a system which is stable under perturbations of the second kind.

在本文中，我们探讨了当受到两种扰动时 Ising Hamiltonian 的能量图谱的稳定性：耦合系数和外部场的扰动，以及底层图结构的扰动。我们给出了充分条件，使得给定哈密顿量的基态在第一类扰动下在顺序保持方面是稳定的。在这里，顺序保持是指对应于扰动哈密顿量中的两个自旋配置的能量顺序将在原始哈密顿量中保持到给定的误差范围。我们还估计了当原始哈密顿量的耦合系数和局部外部磁场是 iid 高斯随机变量时，原始哈密顿量和扰动哈密顿量的基态之间的能隙受给定误差范围限制的概率。最后，我们展示了一个在第二类扰动下稳定的系统的具体例子。