  In this paper, we explore using the Harrow-Hassidim-Lloyd (HHL) algorithm to
address scientific and engineering problems through quantum computing utilizing
the NWQSim simulation package on high-performance computing. Focusing on
domains such as power-grid management and heat transfer problems, we
demonstrate the correlations of the precision of quantum phase estimation,
along with various properties of coefficient matrices, on the final solution
and quantum resource cost in iterative and non-iterative numerical methods such
as Newton-Raphson method and finite difference method, as well as their impacts
on quantum error correction costs using Microsoft Azure Quantum resource
estimator. We conclude the exponential resource cost from quantum phase
estimation before and after quantum error correction and illustrate a potential
way to reduce the demands on physical qubits. This work lays down a preliminary
step for future investigations, urging a closer examination of quantum
algorithms' scalability and efficiency in domain applications.
