{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/variational-quantum-linear-solver-a-hybrid","title":"Variational Quantum Linear Solver","arxiv_id":"1909.05820","date":"2019-09-12","proceeding":null,"authors":["Carlos Bravo-Prieto","Ryan LaRose","M. Cerezo","Yigit Subasi","Lukasz Cincio","Patrick J. Coles"],"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 $|x\\rangle$ such that $A|x\\rangle\\propto|b\\rangle$. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision $\\epsilon$ is achieved. Specifically, we prove that $C \\geq \\epsilon^2 / \\kappa^2$, where $C$ is the VQLS cost function and $\\kappa$ is the condition number of $A$. We present efficient quantum circuits to estimate $C$, 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 $1024\\times1024$. Finally, we numerically solve non-trivial problems of size up to $2^{50}\\times2^{50}$. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in $\\epsilon$, $\\kappa$, and the system size $N$.","url_abs":"https://arxiv.org/abs/1909.05820v4","url_pdf":"https://arxiv.org/pdf/1909.05820v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"variational-quantum-linear-solver-a-hybrid","repo_url":"https://github.com/omalled/quantum-woodbury","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1909.05820","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}