{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:23:45Z","timestamp":1760059425318,"version":"build-2065373602"},"reference-count":53,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,13]],"date-time":"2025-06-13T00:00:00Z","timestamp":1749772800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the National Research Foundation in Singapore"},{"name":"the A*STAR"},{"name":"CQT-Return of PIs EOM YR1-10 Funding"},{"name":"CQT Young Researcher Career Development Grant"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Restricted Boltzmann machines (RBMs) have demonstrated considerable success as variational quantum states; however, their representational power remains incompletely understood. In this work, we present an analytical proof that RBMs can exactly and efficiently represent stabilizer code states\u2014a class of highly entangled quantum states that are central to quantum error correction. Given a set of stabilizer generators, we develop an efficient algorithm to determine both the RBM architecture and the exact values of its parameters. Our findings provide new insights into the expressive power of RBMs, highlighting their capability to encode highly entangled states, and may serve as a useful tool for the classical simulation of quantum error-correcting codes.<\/jats:p>","DOI":"10.3390\/e27060627","type":"journal-article","created":{"date-parts":[[2025,6,13]],"date-time":"2025-06-13T03:53:12Z","timestamp":1749786792000},"page":"627","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["An Efficient Algorithmic Way to Construct Boltzmann Machine Representations for Arbitrary Stabilizer Code"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5415-3307","authenticated-orcid":false,"given":"Yuan-Hang","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Physics, University of California, San Diego, CA 92093, USA"},{"name":"School of the Gifted Young, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8588-173X","authenticated-orcid":false,"given":"Zhian","family":"Jia","sequence":"additional","affiliation":[{"name":"Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore"},{"name":"CAS Key Laboratory of Quantum Information, School of Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Microsoft Station Q and Department of Mathematics, University of California, Santa Barbara, CA 93106, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yu-Chun","family":"Wu","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, School of Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guang-Can","family":"Guo","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, School of Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"022001","DOI":"10.1088\/0034-4885\/75\/2\/022001","article-title":"Hamiltonian complexity","volume":"75","author":"Osborne","year":"2012","journal-title":"Rep. 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