{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:54:17Z","timestamp":1760057657001,"version":"build-2065373602"},"reference-count":66,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,14]],"date-time":"2025-02-14T00:00:00Z","timestamp":1739491200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"PEPR-Q (QubitAF project)","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"International Foundation Big Data and Artificial Intelligence for Human Development (IFAB, project \u201cQuantum Computing for Applications\u201d)","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"INFN (project \u201cQUANTUM\u201d)","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"National Centre for HPC, Big Data, and Quantum Computing","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The use of Neural Networks in quantum many-body theory has undergone a formidable rise in recent years. Among the many possible applications, their pattern recognition power can be utilized when dealing with the study of equilibrium phase diagrams. Learning by Confusion has emerged as an interesting and unbiased scheme within this context. This technique involves systematically reassigning labels to the data in various ways, followed by training and testing the Neural Network. While random labeling results in low accuracy, the method reveals a peak in accuracy when the data are correctly and meaningfully partitioned, even if the correct labeling is initially unknown. Here, we propose a generalization of this confusion scheme for systems with more than two phases, for which it was originally proposed. Our construction relies on the use of a slightly different Neural Network: from a binary classifier, we move to a ternary one, which is more suitable to detect systems exhibiting three phases. After introducing this construction, we test it on free and interacting Kitaev chains and on the one-dimensional Extended Hubbard model, consistently achieving results that are compatible with previous works. Our work opens the way to wider use of Learning by Confusion, demonstrating once more the usefulness of Machine Learning to address quantum many-body problems.<\/jats:p>","DOI":"10.3390\/e27020199","type":"journal-article","created":{"date-parts":[[2025,2,14]],"date-time":"2025-02-14T03:20:52Z","timestamp":1739503252000},"page":"199","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Three-Phase Confusion Learning"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-4498-3680","authenticated-orcid":false,"given":"Filippo","family":"Caleca","sequence":"first","affiliation":[{"name":"Laboratoire de Physique, Centre Nationale de la Recherche Scientifique, \u00c9cole Normale Sup\u00e9rieure de Lyon, Universit\u00e9 Lyon 1, 46 All\u00e9e d\u2019Italie, F-69342 Lyon, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9257-5199","authenticated-orcid":false,"given":"Simone","family":"Tibaldi","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica e Astronomia, Universit\u00e0 di Bologna, Via Irnerio 46, I-40127 Bologna, Italy"},{"name":"Stituto Nazionale di Fisica Nucleare, Sezione di Bologna, I-40127 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6801-5976","authenticated-orcid":false,"given":"Elisa","family":"Ercolessi","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica e Astronomia, Universit\u00e0 di Bologna, Via Irnerio 46, I-40127 Bologna, Italy"},{"name":"Stituto Nazionale di Fisica Nucleare, Sezione di Bologna, I-40127 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Sutherland, B. 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