# Implementation of the HHL Algorithm for Solving the Poisson Equation on Quantum Simulators
## 1 Introduction  
The Poisson equation is one of the fundamental equations of mathematical physics that describes the potential distribution in static fields. The solution of the Poisson equation on a 2D grid can have numerous practical applications in various fields of science and technology, including semiconductor physics, optics, and mechanics, among others. However, conventional methods for solving the Poisson equation on a grid demand significant computational resources and encounter the issue of exponential growth in computational complexity as the grid dimension increases. In this context, new approaches to solving the Poisson equation on quantum computers have emerged, potentially offering advantages over classical methods. One approach is based on the use of quantum computers to solve the Poisson equation on a grid. This approach allows the properties of quantum mechanics, such as the superposition of states, parallel operations, and quantum transformations, to be effectively utilized in solving the problem. In addition, several papers focus on solving the Poisson equation using quantum computers. For example, in ref. [1], the authors focus on the numerical solution of the Poisson equation. The authors specifically propose a quantum method and a scalable quantum scheme that roughly approximates the grid-based solution to the Poisson equation. In ref. [2], the authors discuss various quantum algorithms for solving linear systems of equations, including the Poisson equation. In ref. [3], the gate insulator is investigated using a quality control simulator and IBM equipment, and the authors use quantum computing to solve the Poisson equation. In ref. [4], the authors create a spectral technique for more complex second-order elliptic equations and a quantum finite difference method for the Poisson equation. It is also worth mentioning the article [5], in which the authors propose avariational quantum algorithm (VQA) that can be used with noisy medium-scale quantum computers to solve the Poisson equation. Additionally, there is also the work [6], in which the authors present a quantum algorithm for solving the wave equation, which can be related to the Poisson equation in some applications. In ref. [7], the author describes a quantum algorithm for solving nonlinear differential equations, which, in some cases, can be applied to solve the Poisson equation. In ref. [8], the HHL algorithm is used to solve the Poisson equation for the p-n junction of a semiconductor nanowire at the Neumann boundary, confirming that there is no electric field at the electrode boundaries. It was found that the model of quantum gates proposed by the authors for the implementation of the Neumann boundary condition, in combination with an appropriately executed non-uniform grid, successfully reproduces the result obtained by the traditional method. In ref. [9], the authors present numerous PDE solutions on a quantum computer. Quantum algorithms for differential equations find practical implementation even within these works [10; 11; 12; 13; 14; 15]. To solve the Poisson equation on a 2D grid, one can use the finite difference method. This method consists of approximating the second derivative concerning \(x\) and \(y\) using difference expressions. The result is a system of equations that can be solved numerically. However, this method can be resource intensive and slow when there is a large number of mesh nodes and complex boundary conditions. These are just a few examples of the work that has been done on solving the Poisson equation using quantum computers. However, it should be noted that quantum computing is still at an early stage of development, and the practical application of quantum algorithms to solve real-world problems requires further research and development. In oil production problems, the Poisson equation is used to model fluid flow in a porous medium, determine pressure distribution, and predict fluid flow.