Non-Uniform Quantum Fourier Transform

Quantum Physics arXiv:2602.13472 (quant-ph) [Submitted on 13 Feb 2026] Title:Non-Uniform Quantum Fourier Transform Authors:Junaid Aftab, Yuehaw Khoo, Haizhao Yang View a PDF of the paper titled Non-Uniform Quantum Fourier Transform, by Junaid Aftab and 2 other authors View PDF Abstract:The Discrete Fourier Transform (DFT) is central to the analysis of uniformly sampled signals, yet many practical applications involve non-uniform sampling, requiring the Non-Uniform Discrete Fourier Transform (NUDFT). While quantum algorithms for the standard DFT are well established, a corresponding framework for the non-uniform case remains underdeveloped. This work introduces a quantum algorithm for the Non-Uniform Quantum Fourier Transform (NUQFT) based on a low-rank factorization of the NUDFT matrix. The factorization is translated into an explicit quantum construction using block encodings, Quantum Signal Processing, and the Linear Combination of Unitaries framework, yielding an $\epsilon$-accurate block encoding of the NUDFT matrix with controlled approximation error from both classical truncation and quantum implementation. Under standard oracle access assumptions for non-uniform sampling points, we derive explicit, non-asymptotic gate-level resource estimates. The resulting complexity scales polylogarithmically with target precision, quadratically with the number of qubits through the quantum Fourier transform, and logarithmically with a geometry-dependent conditioning parameter induced by the non-uniform grid. This establishes a concrete and resource-efficient quantum analogue of the NUDFT and provides a foundation for quantum algorithms on irregularly sampled data. Comments: Subjects: Quantum Physics (quant-ph); Numerical Analysis (math.NA) MSC classes: 68Q12, 81P68, 65T50, 15A23 Cite as: arXiv:2602.13472 [quant-ph] (or arXiv:2602.13472v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2602.13472 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Junaid Aftab [view email] [v1] Fri, 13 Feb 2026 21:26:02 UTC (48 KB) Full-text links: Access Paper: View a PDF of the paper titled Non-Uniform Quantum Fourier Transform, by Junaid Aftab and 2 other authorsView PDFTeX Source view license Current browse context: quant-ph new | recent | 2026-02 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?)