# A Quantum Approach for Exploring the Numerical Results of the Heat Equation
## 1 Introduction
Quantum approaches for solving the heat equation are based on discretizing the PDE and reducing the problem to solving a system of linear equations, which are solved mainly using approaches based on the Harrow, Hasidim, and Lloyd (HHL) algorithm [13; 14]. As a rule, a variational quantum linear solver (VQLS) is used for linear systems, based on minimizing a nonlinear cost function, as in [15; 16; 17; 18; 19]. All of these implementations demonstrated an exponential increase in computational speed as the number of qubits increased, which varied from 3 to 11. In [16], three implementations of the above approach were implemented: the direct variational method, the Hadamard testing approach, and the most efficient approach using a tree Ansatz. The VQLS is well analyzed in [17], where the authors propose a hybrid quantum-classical algorithm called a VQLS for solving linear systems on quantum computers of the near future. The authors present efficient quantum schemes for estimating \(C\), which is the cost function in the VQLS, while providing evidence of the time complexity involved in its estimation. Typically, the time complexity of a quantum algorithm depends on the number of qubits in the quantum circuit, but using the Seidel method to divide the complete system of equations into subsystems and their iterative solution allows one to bypass the restrictions on the number of qubits and their connections [20]. In [21], a quantum algorithm was presented for the heat conduction problem by discretizing the PDE. In addition to the above methods, recent advances in the quantum algorithms for dissipative nonlinear differential equations have significantly improved computational efficiency, as in [3], where a quantum algorithm for these equations assuming strong dissipation relative to nonlinear and forcing terms was developed. Utilizing Carleman linearization, it maps nonlinear differential equations to a system of linear equations solved by the quantum linear system algorithm, rendering it suitable for strongly dissipative systems such as certain fluid dynamics and epidemiological models. A review of the literature showed that quantum algorithms for solving partial differential equation problems show an exponential increase in the computation speed with an increasing number of qubits. However, at the same time, there is an exponential increase in the relative error and cost function. Quantum modeling is used to address this problem, evaluating its potential impact on the understanding and methodology for modeling the one-dimensional heat equation. Many previous studies on the heat equation have used quantum algorithms and cost functions [22]. To evaluate the effectiveness of this approach, the probability distribution of the outcomes was calculated and presented as a function of the standard deviation. Thus, this study aims to identify opportunities for using quantum computing in various industries to achieve more accurate and efficient results in thermal conductivity modeling. The quantum algorithm has the potential to provide higher accuracy compared to classical numerical simulation methods and may also be more efficient in solving problems with a large number of parameters. In the context of this study, a quantum approach to the simulation modeling of temperature distribution is presented. This study applies the Qiskit quantum programming toolkit to solve heat conduction problems using quantum gates and an iterative process of heat propagation over time. The purpose of this study is to evaluate the effectiveness and potential of using quantum computing to simulate temperature distribution.