Quantum Kaczmarz Algorithm for Solving Linear Algebraic Equations
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Quantum Physics arXiv:2601.01342 (quant-ph) [Submitted on 4 Jan 2026] Title:Quantum Kaczmarz Algorithm for Solving Linear Algebraic Equations Authors:Nhat A. Nghiem, Tuan K. Do, Trung V. Phan View a PDF of the paper titled Quantum Kaczmarz Algorithm for Solving Linear Algebraic Equations, by Nhat A. Nghiem and 2 other authors View PDF HTML (experimental) Abstract:We introduce a quantum linear system solving algorithm based on the Kaczmarz method, a widely used workhorse for large linear systems and least-squares problems that updates the solution by enforcing one equation at a time. Its simplicity and low memory cost make it a practical choice across data regression, tomographic reconstruction, and optimization. In contrast to many existing quantum linear solvers, our method does not rely on oracle access to query entries, relaxing a key practicality bottleneck. In particular, when the rank of the system of interest is sufficiently small and the rows of the matrix of interest admit an appropriate structure, we achieve circuit complexity $\mathcal{O}\left(\frac{1}{\varepsilon}\log m\right)$, where $m$ is the number of variables and $\varepsilon$ is the target precision, without dependence on the sparsity $s$, and could possibly be without explicit dependence on condition number $\kappa$. This shows a significant improvement over previous quantum linear solvers where the dependence on $\kappa,s$ is at least linear. At the same time, when the rows have an arbitrary structure and have at most $s$ nonzero entries, we obtain the circuit depth $\mathcal{O}\left(\frac{1}{\varepsilon}\log s\right)$ using extra $\mathcal{O}(s)$ ancilla qubits, so the depth grows only logarithmically with sparsity $s$. When the sparsity $s$ grows as $\mathcal{O}(\log m)$, then our method can achieve an exponential improvement with respect to circuit depth compared to existing quantum algorithms, while using (asymptotically) the same amount of qubits. Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2601.01342 [quant-ph] (or arXiv:2601.01342v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2601.01342 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Nhat Anh Vu Nghiem [view email] [v1] Sun, 4 Jan 2026 03:13:36 UTC (29 KB) Full-text links: Access Paper: View a PDF of the paper titled Quantum Kaczmarz Algorithm for Solving Linear Algebraic Equations, by Nhat A. Nghiem and 2 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-01 References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) Links to Code Toggle Papers with Code (What is Papers with Code?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)