# A Quantum Approach for Exploring the Numerical Results of the Heat Equation
## 4 Conclusions  
This article demonstrates the practical application of temperature distribution modeling using quantum computing. A quantum algorithm based on the principles of quantum modeling has been developed, which makes it possible to effectively simulate the evolution of the temperature distribution considering thermal conductivity. Using the statevector and qasm simulations, including in a cloud-based quantum computer, it has been discovered that quantum algorithms can significantly improve the accuracy and efficiency of modeling physical processes such as thermal conductivity. Numerical experiments have revealed a direct relationship between the number of qubits and the accuracy of the simulation. An analysis of various parameters has shown that the time complexity of the quantum algorithm is effectively scaled with the accuracy and size of the equation. The model analysis confirmed good agreement with the analytical solutions, which is confirmed by the low value of the relative error.  
In addition, successful modeling of one-dimensional thermal conductivity problems has confirmed the effectiveness of the proposed method based on quantum gates and states. The results of heuristic scaling showed an almost logarithmic dependence on \(1/\varepsilon\) and a linear dependence on the number of qubits.  
However, it should be noted that in this study, only the small dimensions of the equation were taken into account due to computational limitations. In addition, the optimization of the quantum circuit and the use of an optimizer may be the subject of further research. This opens up new prospects for further research aimed at improving and optimizing methods for solving such problems using quantum computing.  
In the future, it is planned to expand the scope of quantum thermal conductivity modeling methods to larger-scale systems and more complex physical phenomena. Further research may include optimizing algorithms, using hybrid algorithms, increasing the number of qubits to improve accuracy and efficiency, and investigating the effects of various factors on simulation results. This will allow us to better understand the possibilities and limitations of quantum methods in solving problems of thermal conductivity and contribute to the development of this field of science.  
Conceptualization, T.I. and B.D.; methodology, A.M. and N.A.; software, A.M. and N.A.; validation, T.I. and B.D.; formal analysis, A.M., N.A. and B.D.; investigation, T.I. and A.M.; writing--original draft preparation, N.A. and B.D.; writing--review and editing, T.I.; visualization, N.A.; supervision, B.D.; project administration, A.M. All authors have read and agreed to the published version of the manuscript.  
This study was funded by Grant No. AP19576314 \"Development of high-performance hybrid algorithms for solving oil displacement problems based on classical (GPU) and quantum computing\" of the Ministry of Science and Higher Education of the Republic of Kazakhstan.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
Authors [PERSON] and [PERSON] were employed by the company LTD \"DigitAlem\". The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.