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A large class of these models, seen as Fourier series whose frequencies are derived from the encoding gates, were thought to have their Fourier coefficients mostly determined by the trainable gates. Here, we demonstrate a new correlation between the Fourier coefficients of the quantum model and its encoding gates. In addition, we display a phenomenon of vanishing expressivity in certain settings, where some Fourier coefficients vanish exponentially as the number of qubits grows. These two behaviors imply novel forms of constraints which limit the expressivity of PQCs, and therefore imply a new inductive bias for quantum models. The key concept in this work is the notion of a frequency redundancy in the Fourier series spectrum, which determines its importance. Those theoretical behaviors are observed in numerical simulations.<\/jats:p>","DOI":"10.22331\/q-2025-09-03-1847","type":"journal-article","created":{"date-parts":[[2025,9,3]],"date-time":"2025-09-03T10:47:15Z","timestamp":1756896435000},"page":"1847","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":10,"title":["Constrained and Vanishing Expressivity of Quantum Fourier Models"],"prefix":"10.22331","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-3997-2646","authenticated-orcid":false,"given":"Hela","family":"Mhiri","sequence":"first","affiliation":[{"name":"Sorbonne Universit\u00e9, CNRS, LIP6, 75005 Paris, France"},{"name":"ENSTA Paris, Institut Polytechnique de Paris, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-4898-8155","authenticated-orcid":false,"given":"Leo","family":"Monbroussou","sequence":"additional","affiliation":[{"name":"Sorbonne Universit\u00e9, CNRS, LIP6, 75005 Paris, France"},{"name":"CEMIS, Direction Technique, Naval Group, 83190 Ollioules, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0506-7453","authenticated-orcid":false,"given":"Mario","family":"Herrero-Gonzalez","sequence":"additional","affiliation":[{"name":"Sorbonne Universit\u00e9, CNRS, LIP6, 75005 Paris, France"},{"name":"Quantum Software Lab, School of Informatics, University of Edinburgh, United Kingdom"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6291-3927","authenticated-orcid":false,"given":"Slimane","family":"Thabet","sequence":"additional","affiliation":[{"name":"Sorbonne Universit\u00e9, CNRS, LIP6, 75005 Paris, France"},{"name":"PASQAL SAS, 7 avenue L\u00e9onard de Vinci, 91300 Massy, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6280-9604","authenticated-orcid":false,"given":"Elham","family":"Kashefi","sequence":"additional","affiliation":[{"name":"Sorbonne Universit\u00e9, CNRS, LIP6, 75005 Paris, France"},{"name":"Quantum Software Lab, School of Informatics, University of Edinburgh, United Kingdom"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2039-5308","authenticated-orcid":false,"given":"Jonas","family":"Landman","sequence":"additional","affiliation":[{"name":"Quantum Software Lab, School of Informatics, University of Edinburgh, United Kingdom"},{"name":"QC Ware, Palo Alto, USA and Paris, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2025,9,3]]},"reference":[{"key":"0","unstructured":"Nathan Wiebe, Ashish Kapoor, and Krysta M. 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