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We report that a deep learning based simulation can achieve solutions with competitive precision for the spin <jats:inline-formula>\n                     <jats:tex-math\/>\n                     <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi>J<\/mml:mi>\n                        <mml:mn>1<\/mml:mn>\n                     <\/mml:math>\n                     <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"mlstacc56aieqn1.gif\" xlink:type=\"simple\"\/>\n                  <\/jats:inline-formula>\u2013<jats:inline-formula>\n                     <jats:tex-math\/>\n                     <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi>J<\/mml:mi>\n                        <mml:mn>2<\/mml:mn>\n                     <\/mml:math>\n                     <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"mlstacc56aieqn2.gif\" xlink:type=\"simple\"\/>\n                  <\/jats:inline-formula> model and fermionic <jats:italic>t<\/jats:italic>-<jats:italic>J<\/jats:italic> model, on rectangular lattices within periodic boundary conditions. The optimizations of the deep neural networks are performed on the heterogeneous platforms, such as the new generation Sunway supercomputer and the multi graphical-processing-unit clusters. Both high scalability and high performance are achieved within an AI-HPC hybrid framework. The accomplishment of this work opens the door to simulate spin and fermionic lattice models with state-of-the-art lattice size and precision.<\/jats:p>","DOI":"10.1088\/2632-2153\/acc56a","type":"journal-article","created":{"date-parts":[[2023,3,17]],"date-time":"2023-03-17T22:33:20Z","timestamp":1679092400000},"page":"015035","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":13,"title":["Deep learning representations for quantum many-body systems on heterogeneous hardware"],"prefix":"10.1088","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7882-3571","authenticated-orcid":true,"given":"Xiao","family":"Liang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mingfan","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qian","family":"Xiao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Junshi","family":"Chen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chao","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hong","family":"An","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2050-134X","authenticated-orcid":true,"given":"Lixin","family":"He","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"266","published-online":{"date-parts":[[2023,3,30]]},"reference":[{"key":"mlstacc56abib1","doi-asserted-by":"publisher","first-page":"462","DOI":"10.1038\/s42254-021-00324-3","article-title":"High-temperature superconductivity","volume":"3","author":"Zhou","year":"2021","journal-title":"Nat. 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