{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,17]],"date-time":"2026-06-17T02:19:18Z","timestamp":1781662758783,"version":"3.54.5"},"reference-count":72,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,11,22]],"date-time":"2023-11-22T00:00:00Z","timestamp":1700611200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare |<\/mml:mo><\/mml:mrow>x<\/mml:mi>&#x27E9;<\/mml:mo><\/mml:math> such that A<\/mml:mi>|<\/mml:mo><\/mml:mrow>x<\/mml:mi>&#x27E9;<\/mml:mo>&#x221D;<\/mml:mo>|<\/mml:mo><\/mml:mrow>b<\/mml:mi>&#x27E9;<\/mml:mo><\/mml:math>. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision &#x03F5;<\/mml:mi><\/mml:math> is achieved. Specifically, we prove that C<\/mml:mi>&#x2A7E;<\/mml:mo>&#x03F5;<\/mml:mi>2<\/mml:mn><\/mml:msup>\/<\/mml:mo><\/mml:mrow>&#x03BA;<\/mml:mi>2<\/mml:mn><\/mml:msup><\/mml:math>, where C<\/mml:mi><\/mml:math> is the VQLS cost function and &#x03BA;<\/mml:mi><\/mml:math> is the condition number of A<\/mml:mi><\/mml:math>. We present efficient quantum circuits to estimate C<\/mml:mi><\/mml:math>, while providing evidence for the classical hardness of its estimation. Using Rigetti's quantum computer, we successfully implement VQLS up to a problem size of 1024<\/mml:mn>&#x00D7;<\/mml:mo>1024<\/mml:mn><\/mml:math>. Finally, we numerically solve non-trivial problems of size up to 2<\/mml:mn>50<\/mml:mn><\/mml:mrow><\/mml:msup>&#x00D7;<\/mml:mo>2<\/mml:mn>50<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math>. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in &#x03F5;<\/mml:mi><\/mml:math>, &#x03BA;<\/mml:mi><\/mml:math>, and the system size N<\/mml:mi><\/mml:math>.<\/jats:p>","DOI":"10.22331\/q-2023-11-22-1188","type":"journal-article","created":{"date-parts":[[2023,11,22]],"date-time":"2023-11-22T11:22:54Z","timestamp":1700652174000},"page":"1188","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":155,"title":["Variational Quantum Linear Solver"],"prefix":"10.22331","volume":"7","author":[{"given":"Carlos","family":"Bravo-Prieto","sequence":"first","affiliation":[{"name":"Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA."},{"name":"Barcelona Supercomputing Center, Barcelona, Spain."},{"name":"Institut de Ci\u00e8ncies del Cosmos, Universitat de Barcelona, Barcelona, Spain."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ryan","family":"LaRose","sequence":"additional","affiliation":[{"name":"Department of Computational Mathematics, Science, and Engineering & Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48823, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M.","family":"Cerezo","sequence":"additional","affiliation":[{"name":"Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA."},{"name":"Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yigit","family":"Subasi","sequence":"additional","affiliation":[{"name":"Computer, Computational and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lukasz","family":"Cincio","sequence":"additional","affiliation":[{"name":"Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Patrick J.","family":"Coles","sequence":"additional","affiliation":[{"name":"Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2023,11,22]]},"reference":[{"key":"0","doi-asserted-by":"crossref","unstructured":"E. 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