# Parallel Computing of Edwards--Anderson Model
## 1 Introduction  
The Edwards--Anderson model was proposed by [PERSON] and [PERSON] in 1975 to describe the physics of spin glass [1, 2]. The model represents a set of spins arranged at the nodes of the lattice. Each spin can take on a value of \(+1\) or \(-1\) (or the direction up or down in some one dimensional isotopic space). All adjacent spins interact with each other. The interaction energy of one pair is calculated by the formula  
\[E=\sum_{<i,j>}J_{ij}s_{i}s_{j}, \tag{1}\]  
where the sum symbol with the index \(<i,j>\) denotes the summation over all adjacent lattice nodes, \(J_{ij}\) is a coupling constants between spins, taking values \(+1\) or \(-1\) with equal probability for bimodal distribution and any values with probability  
\[P(J_{ij})=\frac{1}{\sqrt{2\pi}\eta_{0}}e^{-\frac{J_{ij}}{2\eta_{0}}} \tag{2}\]  
for Gaussian distribution. Spin glasses have a complex energy landscape and property of ultrametricity [3]. The spin glass model is used in taxonomy, classification problems, information theory, biology, bioinformatics, protein folding and freezing [3], in neural networks, in assessments performance evaluation of quantum computers and in optimisation problems. As neural network theory has evolved, particularly since the creation of the Hopfield network [4, 5, 6], Restricted Boltzmann machine [7] and deep Boltzmann machine [8] interest in spin glasses has increased significantly. In the Hopfield network, finding an image from a distorted input image is reduced to the problem of finding metastable states with maximum correlation of the input image with the trained one. The set of local minima of states corresponds to the set of memorized images. By constructing properly the coupling constants between spins itis possible to control the size of the barriers between states, the number of local minimums, width of valleys, etc. [9] This explains the many publications devoted to the analysis of metastable states and methods for counting their number [10; 11; 12]. Knowing the physics of spin glasses allows us to control performance, error rate and data processing capacity of the neural network. A development of the Hopfield network is the Boltzmann machine, the operation of which has a probabilistic nature. The Boltzmann machine allows us to overcome relatively small local minima and stay at deeper ones. This makes its operation more stable and efficient in practical applications. One important practical application of the ground state search methods is testing quantum computers. Ground state search algorithms can serve as a benchmark for quantum computers. For example, DWave computers are adapted to solve the problem of finding the ground state of spin glasses [13; 14]. Due to the imperfect technology, errors occur in quantum computers that lead to incorrect results. Thus, the problem arises of constructing an a priori known ground state for testing a quantum computer [15]. An interesting approach to tackle this problem is given in [16]. It is based on reducing the problem of solving a system of linear algebraic equations (with algebraic operations modulo two) to the problem of searching for the ground state. It is worth noting that besides quantum computers other analogue devices are under development to solve the optimisation problem of Edwards--Anderson models [15; 16; 17]. The theory of spin glass is closely related to optimization problems, in particular to the problem of the equality of P and NP classes. An exact polynomial-time method for finding the ground state has not yet been found. But there are many probabilistic approaches to solving this problem. We can mention here the Monte Carlo methods [18; 19], simulated annealing algorithms [20; 21], parallel tempering and multicanonical Monte Carlo sampling [22]. It is worth mentioning the exact method based on the Pfaffian calculation and modular arithmetic methods [23]. It is also capable of working with lattices up to size 100 \(\times\) 100.