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

• Published in: Algorithms Vol. 18; no. 8; p. 491
• Main Authors: Zheng, Muqing, Liu, Chenxu, Stein, Samuel, Li, Xiangyu, Mülmenstädt, Johannes, Chen, Yousu, Li, Ang
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.08.2025
• Subjects: Accuracy; Algorithms; Decomposition; Eigenvalues; Error correction; Experiments; Finite difference method; Fourier transforms; hybrid software for QC-HPC; Investigations; Investment analysis; Linear algebra; Newton-Raphson method; Numerical methods; Optimization; Quantum computing; quantum simulation; Qubits (quantum computing); New York; Germany
• ISSN: 1999-4893; 1999-4893
• DOI: 10.3390/a18080491

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 a high-performance computing platform. Focusing on domains such as power-grid management and climate projection, we demonstrate the correlations of the accuracy 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 the Newton–Raphson method and finite difference method, as well as their impacts on quantum error correction costs using the Microsoft Azure Quantum resource estimator. We summarize 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.