{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T00:46:54Z","timestamp":1776732414573,"version":"3.51.2"},"reference-count":24,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2025,3,10]],"date-time":"2025-03-10T00:00:00Z","timestamp":1741564800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>Parametrized quantum circuits (PQC) are quantum circuits which consist of both fixed and parametrized gates. In recent approaches to quantum machine learning (QML), PQCs are essentially ubiquitous and play the role analogous to classical neural networks. They are used to learn various types of data, with an underlying expectation that if the PQC is made sufficiently deep, and the data plentiful, the generalization error will vanish, and the model will capture the essential features of the distribution. While there exist results proving the approximability of square-integrable functions by PQCs under the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi>L<\/mml:mi><mml:mn>2<\/mml:mn><\/mml:msup><\/mml:math> distance, the approximation for other function spaces and under other distances has been less explored. In this work we show that PQCs can approximate the space of continuous functions, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>p<\/mml:mi><\/mml:math>-integrable functions and the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi>H<\/mml:mi><mml:mi>k<\/mml:mi><\/mml:msup><\/mml:math> Sobolev spaces under specific distances. Moreover, we develop generalization bounds that connect different function spaces and distances. These results provide a theoretical basis for different applications of PQCs, for example for solving differential equations. Furthermore, they provide us with new insight on the role of the data normalization in PQCs and of loss functions which better suit the specific needs of the users.<\/jats:p>","DOI":"10.22331\/q-2025-03-10-1658","type":"journal-article","created":{"date-parts":[[2025,3,10]],"date-time":"2025-03-10T14:22:52Z","timestamp":1741616572000},"page":"1658","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":3,"title":["Approximation and Generalization Capacities of Parametrized Quantum Circuits for Functions in Sobolev Spaces"],"prefix":"10.22331","volume":"9","author":[{"given":"Alberto","family":"Manzano","sequence":"first","affiliation":[{"name":"Department of Mathematics and CITIC, Universidade da Coru\u00f1a, Campus de Elvi\u00f1a s\/n, A Coru\u00f1a, Spain"}]},{"given":"David","family":"Dechant","sequence":"additional","affiliation":[{"name":"$\\langle aQa^L\\rangle$ Applied Quantum Algorithms Leiden, The Netherlands"},{"name":"Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands"}]},{"given":"Jordi","family":"Tura","sequence":"additional","affiliation":[{"name":"$\\langle aQa^L\\rangle$ Applied Quantum Algorithms Leiden, The Netherlands"},{"name":"Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands"}]},{"given":"Vedran","family":"Dunjko","sequence":"additional","affiliation":[{"name":"$\\langle aQa^L\\rangle$ Applied Quantum Algorithms Leiden, The Netherlands"},{"name":"LIACS, Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, Netherlands"}]}],"member":"9598","published-online":{"date-parts":[[2025,3,10]]},"reference":[{"key":"0","unstructured":"Robert A Adams and John JF Fournier. Sobolev spaces. Elsevier, 2003."},{"key":"1","doi-asserted-by":"publisher","unstructured":"Victor Burenkov. Extension theorems for sobolev spaces. 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