Two Quantum Algorithms for Nonlinear Reaction-Diffusion Equation using Chebyshev Approximation Method

Quantum Physics arXiv:2510.19855 (quant-ph) [Submitted on 21 Oct 2025] Title:Two Quantum Algorithms for Nonlinear Reaction-Diffusion Equation using Chebyshev Approximation Method Authors:Manish Kumar View a PDF of the paper titled Two Quantum Algorithms for Nonlinear Reaction-Diffusion Equation using Chebyshev Approximation Method, by Manish Kumar View PDF HTML (experimental) Abstract:We present two new quantum algorithms for reaction-diffusion equations that employ the truncated Chebyshev polynomial approximation. This method is employed to numerically solve the ordinary differential equation emerging from the linearization of the associated nonlinear differential equation. In the first algorithm, we use the matrix exponentiation method (Patel et al., 2018), while in the second algorithm, we repurpose the quantum spectral method (Childs et al., 2020). Our main technical contribution is to derive the sufficient conditions for the diagonalization of the Carleman embedding matrix, which is indispensable for designing both quantum algorithms. We supplement this with an efficient iterative algorithm to diagonalize the Carleman matrix. Our first algorithm has gate complexity of O(d$\cdot$log(d)+T$\cdot$polylog(T/$\varepsilon$)). Here $d$ is the size of the Carleman matrix, $T$ is the simulation time, and $\varepsilon$ is the approximation error. The second algorithm is polynomial in $log(d)$, $T$, and $log(1/\varepsilon)$ - the gate complexity scales as O(polylog(d)$\cdot$T$\cdot$polylog(T/$\varepsilon$)). In terms of $T$ and $\varepsilon$, this is comparable to the speedup gained by the current best known quantum algorithm for this problem, the truncated Taylor series method (Costa this http URL., 2025). Our approach has two shortcomings. First, we have not provided an upper bound, in terms of d, on the condition number of the Carleman matrix. Second, the success of the diagonalization is based on a conjecture that a specific trigonometric equation has no integral solution. However, we provide strategies to mitigate these shortcomings in most practical cases. Subjects: Quantum Physics (quant-ph); Pattern Formation and Solitons (nlin.PS); Computational Physics (physics.comp-ph) Cite as: arXiv:2510.19855 [quant-ph] (or arXiv:2510.19855v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2510.19855 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Manish Kumar Mr [view email] [v1] Tue, 21 Oct 2025 19:14:23 UTC (889 KB) Full-text links: Access Paper: View a PDF of the paper titled Two Quantum Algorithms for Nonlinear Reaction-Diffusion Equation using Chebyshev Approximation Method, by Manish KumarView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2025-10 Change to browse by: nlin nlin.PS physics physics.comp-ph 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?)