{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T12:30:28Z","timestamp":1770294628695,"version":"3.49.0"},"reference-count":79,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T00:00:00Z","timestamp":1754438400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Quantum Algorithms and Architecture for Domain Science Initiative (QuAADS)","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]},{"name":"Quantum Algorithms and Architecture for Domain Science Initiative (QuAADS)","award":["DE-AC02-05CH11231"],"award-info":[{"award-number":["DE-AC02-05CH11231"]}]},{"name":"Quantum Algorithms and Architecture for Domain Science Initiative (QuAADS)","award":["DE-AC05-76RL01830"],"award-info":[{"award-number":["DE-AC05-76RL01830"]}]},{"name":"Office of Science of the U.S. Department of Energy","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]},{"name":"Office of Science of the U.S. Department of Energy","award":["DE-AC02-05CH11231"],"award-info":[{"award-number":["DE-AC02-05CH11231"]}]},{"name":"Office of Science of the U.S. Department of Energy","award":["DE-AC05-76RL01830"],"award-info":[{"award-number":["DE-AC05-76RL01830"]}]},{"name":"National Energy Research Scientific Computing Center (NERSC)","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]},{"name":"National Energy Research Scientific Computing Center (NERSC)","award":["DE-AC02-05CH11231"],"award-info":[{"award-number":["DE-AC02-05CH11231"]}]},{"name":"National Energy Research Scientific Computing Center (NERSC)","award":["DE-AC05-76RL01830"],"award-info":[{"award-number":["DE-AC05-76RL01830"]}]},{"name":"Pacific Northwest National Laboratory (PNNL)","award":["DE-AC05-00OR22725"],"award-info":[{"award-number":["DE-AC05-00OR22725"]}]},{"name":"Pacific Northwest National Laboratory (PNNL)","award":["DE-AC02-05CH11231"],"award-info":[{"award-number":["DE-AC02-05CH11231"]}]},{"name":"Pacific Northwest National Laboratory (PNNL)","award":["DE-AC05-76RL01830"],"award-info":[{"award-number":["DE-AC05-76RL01830"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this paper, we explore using the Harrow\u2013Hassidim\u2013Lloyd (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\u2013Raphson 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\u2019 scalability and efficiency in domain applications.<\/jats:p>","DOI":"10.3390\/a18080491","type":"journal-article","created":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T15:09:53Z","timestamp":1754492993000},"page":"491","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["An Early Investigation of the HHL Quantum Linear Solver for Scientific Applications"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6659-9672","authenticated-orcid":false,"given":"Muqing","family":"Zheng","sequence":"first","affiliation":[{"name":"Pacific Northwest National Laboratory, Richland, WA 99354, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2616-3126","authenticated-orcid":false,"given":"Chenxu","family":"Liu","sequence":"additional","affiliation":[{"name":"Pacific Northwest National Laboratory, Richland, WA 99354, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2655-8251","authenticated-orcid":false,"given":"Samuel","family":"Stein","sequence":"additional","affiliation":[{"name":"Pacific Northwest National Laboratory, Richland, WA 99354, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5722-0018","authenticated-orcid":false,"given":"Xiangyu","family":"Li","sequence":"additional","affiliation":[{"name":"Pacific Northwest National Laboratory, Richland, WA 99354, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1105-6678","authenticated-orcid":false,"given":"Johannes","family":"M\u00fclmenst\u00e4dt","sequence":"additional","affiliation":[{"name":"Pacific Northwest National Laboratory, Richland, WA 99354, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7591-9597","authenticated-orcid":false,"given":"Yousu","family":"Chen","sequence":"additional","affiliation":[{"name":"Pacific Northwest National Laboratory, Richland, WA 99354, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3734-9137","authenticated-orcid":false,"given":"Ang","family":"Li","sequence":"additional","affiliation":[{"name":"Pacific Northwest National Laboratory, Richland, WA 99354, USA"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,6]]},"reference":[{"key":"ref_1","first-page":"553","article-title":"Rapid solution of problems by quantum computation","volume":"439","author":"Deutsch","year":"1992","journal-title":"Proc. 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