Probabilistic modeling over permutations using quantum computers
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Quantum Physics arXiv:2603.22401 (quant-ph) [Submitted on 23 Mar 2026] Title:Probabilistic modeling over permutations using quantum computers Authors:Vasilis Belis, Giulio Crognaletti, Matteo Argenton, Michele Grossi, Maria Schuld View a PDF of the paper titled Probabilistic modeling over permutations using quantum computers, by Vasilis Belis and 4 other authors View PDF HTML (experimental) Abstract:Quantum computers provide a super-exponential speedup for performing a Fourier transform over the symmetric group, an ability for which practical use cases have remained elusive so far. In this work, we leverage this ability to unlock spectral methods for machine learning over permutation-structured data, which appear in applications such as multi-object tracking and recommendation systems. It has been shown previously that a powerful way of building probabilistic models over permutations is to use the framework of non-Abelian harmonic analysis, as the model's group Fourier spectrum captures the interaction complexity: "low frequencies" correspond to low order correlations, and "high frequencies" to more complex ones. This can be used to construct a Markov chain model driven by alternating steps of diffusion (a group-equivariant convolution) and conditioning (a Bayesian update). However, this approach is computationally challenging and hence limited to simple approximations. Here we construct a quantum algorithm that encodes the exact probabilistic model -- a classically intractable object -- into the amplitudes of a quantum state by making use of the Quantum Fourier Transform (QFT) over the symmetric group. We discuss the scaling, limitations, and practical use of such an approach, which we envision to be a first step towards useful applications of non-Abelian QFTs. Comments: Subjects: Quantum Physics (quant-ph); Machine Learning (cs.LG) Cite as: arXiv:2603.22401 [quant-ph] (or arXiv:2603.22401v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2603.22401 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Vasilis Belis [view email] [v1] Mon, 23 Mar 2026 18:00:12 UTC (605 KB) Full-text links: Access Paper: View a PDF of the paper titled Probabilistic modeling over permutations using quantum computers, by Vasilis Belis and 4 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-03 Change to browse by: cs cs.LG 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?)