Title: An Early Investigation of the HHL Quantum Linear Solver for Scientific Applications

Abstract: 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.