Stability of energy landscape for Ising models

Finding optimal solutions for combinatorial optimization problems, some of which are known to be NP-hard, is a very important problem. Among many possible approaches to such problems, the application of Ising models to solve real social problems has been getting attention due to its versatility (see [1]). More precisely, a given social combinatorial optimization problem can be mapped into a Hamiltonian $H$ on a graph $G=(V,E)$, whose expression is given by

for every Ising spin configuration $\sigma\in\{-1,1\}^{V}$, where $\{J_{b}\}_{b\in E}$ are coupling coefficients and $\{h_{x}\}_{x\in V}$ are local magnetic fields. In that approach, an optimal solution for the intended combinatorial problem corresponds to a ground state (or global minimum) $\sigma_{G}$ of $H$, that is, $\sigma_{G}\in\mathrm{arg\,min}\,H$. There are some well-known methods that can be applied to obtain a ground state. Implementing a Markov chain Monte Carlo (such as Glauber dynamics and stochastic cellular automata) is known as a way to find an approximation for the Gibbs distribution whose highest peaks correspond to the ground states of $H$. We refer for details to [2, 3, 4, 5] and also [6].

However, as long as we use Ising machines or any computer to perform numerical simulations to find a ground state, we cannot avoid the error occurring due to the analog nature or the difficulty of representing real numbers (see [7]). Because of these reasons, we should incorporate the error coming from the coupling coefficients and local magnetic fields by introducing a perturbed Hamiltonian. Hence, our original Hamiltonian $H$ will be perturbed, originating a perturbed Hamiltonian $H_{\delta}$ whose coupling coefficients and local magnetic fields have a maximal error $\delta$. Then, the following natural questions arise:

1. (1)

   For any pair of configurations which are ordered in terms of energy with respect to the perturbed Hamiltonian $H_{\delta}$, is that ordering preserved in the original Hamiltonian $H$, up to a given error margin?

2. (2)

   Given a Hamiltonian $H$ with coupling coefficients and local magnetic fields distributed as i.i.d. Gaussian random variables, what is the probability that the energy gap in $H$ between two ground states respectively for $H$ and the corresponding perturbed Hamiltonian $H_{\delta}$ is sufficiently small?

In addition to the above questions (1) and (2), the following problem is also important when using Ising machines and computers. It may be somewhat a waste of resources taking all coupling coefficients and local magnetic fields into account. It may be useful to “eliminate” vertices of a given graph whose contribution to the total energy is relatively small, in order to save memories of computers. Hence, we also have the following natural question:

1. (3)

   Can we find a subset of a given graph such that for an arbitrary choice of configuration outside of that region, the energy variation can be controlled?

In this paper, we investigate the stability of energy landscape of a given Hamiltonian under perturbations from the view point of order preservation, aiming at answering the questions we addressed above. Thanks to the order preservation property, we can obtain better estimates for the success probability of finding a ground state compared to the result given in [7].

This paper is organized as follows. In Section 2, we provide a precise formulation for the questions we just posed and raise them again. In Section 3, we answer the questions (1’) and (2’) from Section 2. In Section 4, we provide an example together with a sufficient condition that guarantees a positive answer for question (3’).

In this section, we introduce some necessary definitions and terminologies for discussing the stability of energy landscape. Further we also introduce the notion of order preservation for a perturbed system, which plays a central role in this paper. Here, order preservation means, roughly speaking, if we take a ground state for a perturbed Hamiltonian (implemented by a device) then it should be close to the ground state for an original Hamiltonian (intended mathematical problem) in energy, up to a given error margin.

Let us begin by introducing the precise setting. Let $G=(V,E)$ be a finite simple graph with the vertex set $V$ and the edge set $E$. The so-called original Hamiltonian $H$ with coupling coefficients $\{J_{b}\}_{b\in E}$ and external magnetic fields $\{h_{x}\}_{x\in V}$ on $G$ is defined by

for each $\sigma=\{\sigma_{x}\}_{x\in V}\in\{-1,1\}^{V}$. Such a function $H$ can be regarded as a cost function of an intended problem. Given $\delta>0$, we denote by $H_{\delta}$ the perturbed Hamiltonian with the coupling coefficients $\{J^{\prime}_{b}\}_{b\in E}$ and external fields $\{h^{\prime}_{x}\}_{x\in V}$, i.e.,

where the $J^{\prime}_{b}$’s and $h^{\prime}_{x}$’s satisfy the bounds $\sup_{b}\lvert J_{b}-J^{\prime}_{b}\rvert\leq\delta$ and $\sup_{x}\lvert h_{x}-h^{\prime}_{x}\rvert\leq\delta$. This perturbation will be often interpreted as a round-off in the following way. Let $(J_{b}^{(1)}J_{b}^{(2)}\dots)$ and $(h_{x}^{(1)}h_{x}^{(2)}\dots)$ be the binary expansions of the fractional parts of $J_{b}$ and $h_{x}$, i.e.,

where $J_{b}^{(0)},h_{x}^{(0)}\in\mathbb{Z}$ and $J_{b}^{(i)},h_{x}^{(i)}\in\{0,1\}$ for $i\geq 1$. If we set $J_{b}^{\prime}=J_{b}^{(0)}+\sum_{i=1}^{N}\frac{J_{b}^{(i)}}{2^{i}}$ and $h_{x}^{\prime}=h_{x}^{(0)}+\sum_{i=1}^{N}\frac{h_{x}^{(i)}}{2^{i}}$ in the equation (2.2), then the error $\delta$ can be taken as $2^{-N}$. It means that the perturbed Hamiltonian $H_{\delta}$ is obtained by rounding off the given parameters $J_{b}$’s and $h_{x}$’s uniformly from the $(N+1)$-th digit of their binary expansions.

The main purpose of this paper is to clarify the stability of the ground states for a given Hamiltonian under a perturbation in terms of order preservation. In this paper, we will answer the following questions:

1. (1’)

   Find a $\delta>0$ corresponding to a given $\varepsilon>0$, so that, for any pair $(\sigma,\tau)$ that satisfies $H_{\delta}(\sigma)\geq H_{\delta}(\tau)$, the ordering is preserved in $H$ up to the error margin $\varepsilon\sup_{\xi,\eta}\left\lvert H(\xi)-H(\eta)\right\rvert$, i.e.,

   $\displaystyle H(\sigma)\geq H(\tau)-\varepsilon\sup_{\xi,\eta}\left\lvert H(\xi)-H(\eta)\right\rvert.$

   (2.4)

   Here, $\sup_{\xi,\eta}\left\lvert H(\xi)-H(\eta)\right\rvert$ is the total margin of the original Hamiltonian.

2. (2’)

   Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and let $\{J_{b}\}_{b\in E}$ and $\{h_{x}\}_{x\in V}$ be mutually independent standard Gaussian random variables on this probability space. Estimate the probability that the energy gap in $H$ between ground states for $H$ and $H_{\delta}$, say $\sigma_{G}$ and $\tilde{\sigma}_{G}$, respectively, is controled by the given error margin, explicitly,

   $\displaystyle\mathbb{P}\left(0\leq H(\tilde{\sigma}_{G})-H(\sigma_{G})\leq\varepsilon\sup_{\xi,\eta}\left\lvert H(\xi)-H(\eta)\right\rvert\right).$

   (2.5)

A different aspect of stability of a given system is to find a nontrivial subsystem so that the energy gap between any two spin configurations whose spins restricted to the subgraph coincide is bounded above by a given error margin. Also, at the same time, we require that the number of vertices that can be disregarded is at least of order $N^{\alpha}$, where $N=\lvert V\rvert$ and $\alpha\in{[}0,1)$, so that such a number can go to infinity as $N\to\infty$. In the later part of this paper, we answer the following question for a particular case:

1. (3’)

   Let $\lvert V\rvert=N$, and let $\{J_{b}\}_{b\in E}$ and $\{h_{x}\}_{x\in V}$ be mutually independent standard Gaussian random variables. Find a subset $V_{0}\subset V$ for a given $\varepsilon>0$ and $\alpha\in{[}0,1)$ such that

   $\displaystyle{\mathbb{P}}\left(\sup_{\sigma,\tau\in\{-1,1\}^{N}}\left\lvert H(\sigma)-H(\sigma_{V_{0}},\tau_{V\setminus V_{0}})\right\rvert<\varepsilon\sup_{\xi,\eta}\left\lvert H(\xi)-H(\eta)\right\rvert\;\&\;CN^{\alpha}\leq\lvert V\setminus V_{0}\rvert<N\right)$

   is close to $1$, where $\sigma_{V_{0}}\in\{-1,1\}^{V_{0}}$ is the spin configuration $\sigma$ restricted to $V_{0}$ and $\tau_{V\setminus V_{0}}\in\{-1,1\}^{V\setminus V_{0}}$ is the restriction of the spin configuration $\tau$ to $V\setminus V_{0}$.

Questions (1’), (2’) and (3’) above correspond to questions (1), (2) and (3) from Section 1, respectively. In Section 3, we investigate the first two questions above, where for the second one we adopt two different approaches. We obtain answers for question (2’) by means of a method involving the $L^{\infty}$-distance and a graph’s structure approach, and we compare these two methods for three different graphs. Specifically, we consider sufficient conditions on the perturbation $\delta$ to satisfy order preservation, and calculate the probability that such a sufficient condition holds. In Section 4, we obtain an answer for the question (3’) when the graph is a one-dimensional torus $\mathbb{Z}/N\mathbb{Z}$ without external fields.

This part is dedicated to provide solutions for questions (1’) and (2’) just posed in the end of the previous section. Before we proceed to the next sections, let us introduce the quantity $R_{H}$ defined by

which is defined whenever a Hamiltonian $H$ is given. Moreover, if $G=(V,E)$ is a finite simple graph, then we define $k_{G}$ by

Keeping in mind the mathematical setting introduced in the beginning of Section 2, let us start by showing that the order preservation property holds, that is, let us first answer the question (1’), which consists in finding a $\delta>0$ corresponding to a given $\varepsilon>0$ such that $H_{\delta}(\sigma)\geq H_{\delta}(\tau)$ implies $H(\sigma)\geq H(\tau)-\varepsilon R_{H}$; and later on, assuming some randomness on the spin-spin couplings and external fields, we adopt two different approaches to answer question (2’) and estimate the probability that the condition $H(\tilde{\sigma}_{G})-H(\sigma_{G})\leq\varepsilon R_{H}$ is satisfied.

In order to solve the second problem, we will adopt two distinct approaches: a method that relies on uniform estimates and a method where combinatorial estimates are considered instead, which will be presented in Sections 3.2 and 3.3, respectively. In the last part of this section, we compare these two methods and conclude that depending on the underlying graph structure of the problem, one of them will give us a better lower bound for the probability from equation (2.5).

In this section, we consider the stability of energy landscape when a given spin system defined on a graph is compressed into a smaller subsystem. Differently from the previous sections, we fix a given Hamiltonian and we assume a sufficient condition that guarantees the existence of a nontrivial subset of the entire vertex set outside of which we can randomly assign any spin configuration and the energy of the system is kept under control up to a certain error margin.

Let $G=(V,E)$ be a finite simple graph, and let $H$ be the Hamiltonian on $G$ given by

for every configuration $\sigma\in\{-1,1\}^{V}$, where $\{J_{x,y}\}_{\{x,y\}\in E}$ is a collection of mutually independent standard Gaussian random variables. What we would like to show is that we can compress the whole system into a nontrivial subsystem so that the energy landscape of such subsystem is close to the original one up to a given error margin. More precisely, our goal is to find a class of examples for which given a positive constant $\varepsilon$, there is a positive $\delta$ such that the subsystem $V_{0}=V_{0}(\delta)$ of $V$, defined from the relation

is non-trivial, has size comparable to the size of $V$, and satisfies

with high probability, see Figure 2.