Quantum Computing Solution to Sturm-Liouville Differential Equation
International Journal of Innovative Research in Physics · 2024
Quantum computing is coming up as futuristic technology in engineering and science.The famous Sturm-Liouville (SL) differential equation along with some specific boundary conditions appear in many branches of physics as well as in other fundamental fields.The numerical solution of such boundary value problems (like heat transfer, potential problem etc.) is frequently computationally expensive, and traditional computers may not be able to deliver accurate solutions in an acceptable length of time.We propose quantum computing approach to solve such boundary value problems that will be more efficient and time solvent.The present research demands that the Harrow-Hassidim-Lloyd (HHL) quantum algorithm provides an efficient technique for solving such boundary value problems that arise in many branches of physical science and engineering.The prime focus of this research is to explore HHL quantum algorithm in detail.The quantum phase estimation algorithm is applied for solving eigen values and eigenvectors associated with the SL type of boundary value problem.As a case study, in this research, we have solved the steady state heat transfer problem with boundary conditions.Our computation is based on Qiskit module made by IBM and available as a library in Python.A detailed analysis of the computational complexity of our algorithm that compares its performance with classical methods on a range of boundary value problems has been discussed.Role of Tridiagonal Toeplitz matrix is revealed.Results demonstrate that the method of quantum algorithm for solving boundary value problems can significantly outperform classical methods, especially for larger problem sizes.This work represents a significant methodology in solving important equations of physics and mathematics in future quantum computers.