Shock-Capturing Quantum Algorithm for the Linear Advection Operator

Quantum Physics arXiv:2609.21153 (quant-ph) [Submitted on 17 Sep 2026] Title:Shock-Capturing Quantum Algorithm for the Linear Advection Operator Authors:Samuel Hagele, William Gregory, Yuan Shi View a PDF of the paper titled Shock-Capturing Quantum Algorithm for the Linear Advection Operator, by Samuel Hagele and 2 other authors View PDF Abstract:The linear advection operator is an ubiquitous building block in fluid and plasma problems. We develop a quantum algorithm for enacting the operator. When the advection velocity is constant in space, our algorithm is exponentially more efficient per time step than classical and avoids spurious oscillations near steep gradients. The algorithm is most cleanly illustrated using the one-dimensional advection equation on a uniform spatial grid with periodic boundary conditions, which can be extended to higher dimensions. The algorithm uses a first-order upwind scheme, which captures discontinuities in the wave envelope but is not unitary. We embed the non-unitary upwind scheme using Linear Combinations of Unitaries (LCUs), and develop an efficient quantum gate decomposition of the upwind unitary, which performs one step of advection using $O(n^2)$ two-qubit gates, where $N=2^n$ is the number of spatial grid points, as opposed to a classical computer which costs $O(N)$. Although LCUs introduces a small bounded probability of failure per time step, we show that the accumulation of failures does not lead to exponential-in-time complexity as one would naively expect. Moreover, when LCUs fails, we develop a probabilistic scheme to recover from the failure state, which avoids a full restart of the simulation. The failure recovery scheme uses quantum Fourier transform (QFT) and effectively achieves quantum indefinite integration of an unknown quantum state. The recovery, which can itself fail, is more efficient than a full restart if the wave envelop is well-resolved to include only low Fourier modes. We emulate our scheme classically and demonstrate small problems on Quantinuum's trapped-ion qubits. Our quantum algorithm provides a subroutine for physics simulations that involve linear advection. Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2609.21153 [quant-ph] (or arXiv:2609.21153v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2609.21153 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Samuel Hagele [view email] [v1] Thu, 17 Sep 2026 23:46:14 UTC (1,970 KB) Full-text links: Access Paper: View a PDF of the paper titled Shock-Capturing Quantum Algorithm for the Linear Advection Operator, by Samuel Hagele and 2 other authorsView PDFTeX Source view license Current browse context: quant-ph new | recent | 2026-09 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?) 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?)