A Linear Combination of Unitaries Decomposition for the Laplace Operator
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Quantum Physics arXiv:2601.06370 (quant-ph) [Submitted on 10 Jan 2026] Title:A Linear Combination of Unitaries Decomposition for the Laplace Operator Authors:Thomas Hogancamp, Reuben Demirdjian, Daniel Gunlycke View a PDF of the paper titled A Linear Combination of Unitaries Decomposition for the Laplace Operator, by Thomas Hogancamp and 2 other authors View PDF HTML (experimental) Abstract:We provide novel linear combination of unitaries decompositions for a class of discrete elliptic differential operators. Specifically, Poisson problems augmented with periodic, Dirichlet, Neumann, Robin, and mixed boundary conditions are considered on the unit interval and on higher-dimensional rectangular domains. The number of unitary terms required for our decomposition is independent of the number of grid points used in the discretization and scales linearly with the spatial dimension. Explicit circuit constructions for each unitary are given and their complexities analyzed. The worst case depth and elementary gate cost of any such circuit is shown to scale at most logarithmically with respect to number of grid points in the underlying discrete system. We also investigate the cost of using our method within the Variational Quantum Linear Solver algorithm and show favorable scaling. Finally, we extend the proposed decomposition technique to treat problems that include first-order derivative terms with variable coefficients. Comments: Subjects: Quantum Physics (quant-ph); Numerical Analysis (math.NA) Cite as: arXiv:2601.06370 [quant-ph] (or arXiv:2601.06370v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2601.06370 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Thomas Hogancamp [view email] [v1] Sat, 10 Jan 2026 00:54:39 UTC (144 KB) Full-text links: Access Paper: View a PDF of the paper titled A Linear Combination of Unitaries Decomposition for the Laplace Operator, by Thomas Hogancamp and 2 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-01 Change to browse by: cs cs.NA math math.NA 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?)