Achieving double-logarithmic precision dependence in optimization-based quantum unstructured search
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Quantum Physics arXiv:2603.26039 (quant-ph) [Submitted on 27 Mar 2026] Title:Achieving double-logarithmic precision dependence in optimization-based quantum unstructured search Authors:Zhijian Lai, Dong An, Jiang Hu, Zaiwen Wen View a PDF of the paper titled Achieving double-logarithmic precision dependence in optimization-based quantum unstructured search, by Zhijian Lai and 3 other authors View PDF HTML (experimental) Abstract:Grover's algorithm is a fundamental quantum algorithm that achieves a quadratic speedup for unstructured search problems of size $N$. Recent studies have reformulated this task as a maximization problem on the unitary manifold and solved it via linearly convergent Riemannian gradient ascent (RGA) methods, resulting in a complexity of $O(\sqrt{N}\log (1/\varepsilon))$. In this work, we adopt the Riemannian modified Newton (RMN) method to solve the quantum search problem. We show that, in the setting of quantum search, the Riemannian Newton direction is collinear with the Riemannian gradient in the sense that the Riemannian gradient is always an eigenvector of the corresponding Riemannian Hessian. As a result, without additional overhead, the proposed RMN method numerically achieves a quadratic convergence rate with respect to error $\varepsilon$, implying a complexity of $O(\sqrt{N}\log\log (1/\varepsilon))$, which is double-logarithmic in precision. Furthermore, our approach remains Grover-compatible, namely, it relies exclusively on the standard Grover oracle and diffusion operators to ensure algorithmic implementability, and its parameter update process can be efficiently precomputed on classical computers. Comments: Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optimization and Control (math.OC) MSC classes: 81P68, 90C26, 65K10 ACM classes: F.2.2; G.1.6 Cite as: arXiv:2603.26039 [quant-ph] (or arXiv:2603.26039v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2603.26039 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Zhijian Lai [view email] [v1] Fri, 27 Mar 2026 03:19:27 UTC (93 KB) Full-text links: Access Paper: View a PDF of the paper titled Achieving double-logarithmic precision dependence in optimization-based quantum unstructured search, by Zhijian Lai and 3 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-03 Change to browse by: math math-ph math.MP math.OC 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?)