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We use an extension of the covariance matrix formalism to efficiently track relative phases in the superpositions of Gaussian states. We get an exact simulation algorithm, which costs quadratically with the number of Gaussian states required to represent the initial state, and an approximate simulation algorithm, which costs linearly with the\n \n \n l<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n norm of the coefficients associated with the superposition. We define measures of non-Gaussianity quantifying this simulation cost, which we call the Gaussian rank and the Gaussian extent. From the perspective of quantum resource theories, we investigate the properties of this type of non-Gaussianity measure and compute optimal decompositions for states relevant to continuous-variable quantum computing.\n <\/jats:p>","DOI":"10.22331\/q-2025-10-13-1881","type":"journal-article","created":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T08:58:13Z","timestamp":1760345893000},"page":"1881","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":["Classical simulation and quantum resource theory of non-Gaussian optics"],"prefix":"10.22331","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1677-8696","authenticated-orcid":false,"given":"Oliver","family":"Hahn","sequence":"first","affiliation":[{"name":"Wallenberg Centre for Quantum Technology, Department of Microtechnology and Nanoscience, Chalmers University of Technology, Sweden , SE-412 96 G\u00f6teborg, Sweden"},{"name":"Department of Basic Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8902, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3837-8159","authenticated-orcid":false,"given":"Ryuji","family":"Takagi","sequence":"additional","affiliation":[{"name":"Department of Basic Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8902, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7130-6723","authenticated-orcid":false,"given":"Giulia","family":"Ferrini","sequence":"additional","affiliation":[{"name":"Wallenberg Centre for Quantum Technology, Department of Microtechnology and Nanoscience, Chalmers University of Technology, Sweden , SE-412 96 G\u00f6teborg, Sweden"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3521-831X","authenticated-orcid":false,"given":"Hayata","family":"Yamasaki","sequence":"additional","affiliation":[{"name":"Department of Physics, Graduate School of Science, The University of Tokyo, 7\u20133\u20131 Hongo, Bunkyo-ku, Tokyo, 113\u20130033, Japan"},{"name":"Department of Computer Science, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2025,10,13]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Nissim Ofek, Andrei Petrenko, Reinier Heeres, Philip Reinhold, Zaki Leghtas, Brian Vlastakis, Yehan Liu, Luigi Frunzio, S. 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