{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T04:36:45Z","timestamp":1779251805454,"version":"3.51.4"},"reference-count":97,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,9,30]],"date-time":"2021-09-30T00:00:00Z","timestamp":1632960000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Generative modelling is an important unsupervised task in machine learning. In this work, we study a hybrid quantum-classical approach to this task, based on the use of a quantum circuit born machine. In particular, we consider training a quantum circuit born machine using f-divergences. We first discuss the adversarial framework for generative modelling, which enables the estimation of any f-divergence in the near term. Based on this capability, we introduce two heuristics which demonstrably improve the training of the born machine. The first is based on f-divergence switching during training. The second introduces locality to the divergence, a strategy which has proved important in similar applications in terms of mitigating barren plateaus. Finally, we discuss the long-term implications of quantum devices for computing f-divergences, including algorithms which provide quadratic speedups to their estimation. In particular, we generalise existing algorithms for estimating the Kullback\u2013Leibler divergence and the total variation distance to obtain a fault-tolerant quantum algorithm for estimating another f-divergence, namely, the Pearson divergence.<\/jats:p>","DOI":"10.3390\/e23101281","type":"journal-article","created":{"date-parts":[[2021,9,30]],"date-time":"2021-09-30T10:22:42Z","timestamp":1632997362000},"page":"1281","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6577-3926","authenticated-orcid":false,"given":"Chiara","family":"Leadbeater","sequence":"first","affiliation":[{"name":"Cambridge Quantum Computing Limited, London SW1E 6DR, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1691-1215","authenticated-orcid":false,"given":"Louis","family":"Sharrock","sequence":"additional","affiliation":[{"name":"Cambridge Quantum Computing Limited, London SW1E 6DR, UK"},{"name":"Department of Mathematics, Imperial College London, London SW7 2AZ, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3436-8458","authenticated-orcid":false,"given":"Brian","family":"Coyle","sequence":"additional","affiliation":[{"name":"Cambridge Quantum Computing Limited, London SW1E 6DR, UK"},{"name":"School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0231-1729","authenticated-orcid":false,"given":"Marcello","family":"Benedetti","sequence":"additional","affiliation":[{"name":"Cambridge Quantum Computing Limited, London SW1E 6DR, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"23023","DOI":"10.1088\/1367-2630\/18\/2\/023023","article-title":"The theory of variational hybrid quantum-classical algorithms","volume":"18","author":"McClean","year":"2016","journal-title":"New J. 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