Optimal Scaling Quantum Interior Point Method for Linear Optimization

Quantum Physics arXiv:2512.04510 (quant-ph) [Submitted on 4 Dec 2025] Title:Optimal Scaling Quantum Interior Point Method for Linear Optimization Authors:Mohammadhossein Mohammadisiahroudi, Zeguan Wu, Pouya Sampourmahani, Jun-Kai You, Tamás Terlaky View a PDF of the paper titled Optimal Scaling Quantum Interior Point Method for Linear Optimization, by Mohammadhossein Mohammadisiahroudi and 4 other authors View PDF HTML (experimental) Abstract:The emergence of huge-scale, data-intensive linear optimization (LO) problems in applications such as machine learning has driven the need for more computationally efficient interior point methods (IPMs). While conventional IPMs are polynomial-time algorithms with rapid convergence, their per-iteration cost can be prohibitively high for dense large-scale LO problems. Quantum linear system solvers have shown potential in accelerating the solution of linear systems arising in IPMs. In this work, we introduce a novel almost-exact quantum IPM, where the Newton system is constructed and solved on a quantum computer, while solution updates occur on a classical machine. Additionally, all matrix-vector products are performed on the quantum hardware. This hybrid quantum-classical framework achieves an optimal worst-case scaling of $\mathcal{O}(n^2)$ for fully dense LO problems. To ensure high precision, despite the limited accuracy of quantum operations, we incorporate iterative refinement techniques both within and outside the proposed IPM iterations. The proposed algorithm has a quantum complexity of $\mathcal{O}(n^{1.5} \kappa_A \log(\frac{1}{\epsilon}))$ queries to QRAM and $\mathcal{O}(n^2 \log(\frac{1}{\epsilon}))$ classical arithmetic operations. Our method outperforms the worst-case complexity of prior classical and quantum IPMs, offering a significant improvement in scalability and computational efficiency. Comments: Subjects: Quantum Physics (quant-ph) MSC classes: 90C51, 90C05, 81P68 Cite as: arXiv:2512.04510 [quant-ph] (or arXiv:2512.04510v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2512.04510 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Journal reference: M. Mohammadisiahroudi, Z. Wu, P. Sampourmahani, J. -K. You and T. Terlaky, "Optimal Scaling Quantum Interior Point Method for Linear Optimization," 2025 IEEE QCE, Albuquerque, NM, USA, 2025, pp. 320-326 Related DOI: https://doi.org/10.1109/QCE65121.2025.00044. Focus to learn more DOI(s) linking to related resources Submission history From: Pouya Sampourmahani [view email] [v1] Thu, 4 Dec 2025 06:44:22 UTC (131 KB) Full-text links: Access Paper: View a PDF of the paper titled Optimal Scaling Quantum Interior Point Method for Linear Optimization, by Mohammadhossein Mohammadisiahroudi and 4 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2025-12 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?)