{"id":101723,"date":"2025-05-14T20:43:12","date_gmt":"2025-05-14T20:43:12","guid":{"rendered":"https:\/\/www.europesays.com\/uk\/101723\/"},"modified":"2025-05-14T20:43:12","modified_gmt":"2025-05-14T20:43:12","slug":"quantum-error-correction-of-qudits-beyond-break-even","status":"publish","type":"post","link":"https:\/\/www.europesays.com\/uk\/101723\/","title":{"rendered":"Quantum error correction of qudits beyond break-even"},"content":{"rendered":"<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"1.\">\n<p class=\"c-article-references__text\" id=\"ref-CR1\">Blume-Kohout, R., Caves, C. M. &amp; Deutsch, I. H. Climbing mount scalable: physical resource requirements for a scalable quantum computer. Found. Phys. <b>32<\/b>, 1641\u20131670 (2002).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1023\/A:1021471621587\" data-track-item_id=\"10.1023\/A:1021471621587\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1023%2FA%3A1021471621587\" aria-label=\"Article reference 1\" data-doi=\"10.1023\/A:1021471621587\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=1954911\" aria-label=\"MathSciNet reference 1\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 1\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Climbing%20mount%20scalable%3A%20physical%20resource%20requirements%20for%20a%20scalable%20quantum%20computer&amp;journal=Found.%20Phys.&amp;doi=10.1023%2FA%3A1021471621587&amp;volume=32&amp;pages=1641-1670&amp;publication_year=2002&amp;author=Blume-Kohout%2CR&amp;author=Caves%2CCM&amp;author=Deutsch%2CIH\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"2.\">\n<p class=\"c-article-references__text\" id=\"ref-CR2\">Greentree, A. D. et al. Maximizing the Hilbert space for a finite number of distinguishable quantum states. Phys. Rev. Lett. <b>92<\/b>, 097901 (2004).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevLett.92.097901\" data-track-item_id=\"10.1103\/PhysRevLett.92.097901\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevLett.92.097901\" aria-label=\"Article reference 2\" data-doi=\"10.1103\/PhysRevLett.92.097901\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2004PhRvL..92i7901G\" aria-label=\"ADS reference 2\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=15089514\" aria-label=\"PubMed reference 2\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 2\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Maximizing%20the%20Hilbert%20space%20for%20a%20finite%20number%20of%20distinguishable%20quantum%20states&amp;journal=Phys.%20Rev.%20Lett.&amp;doi=10.1103%2FPhysRevLett.92.097901&amp;volume=92&amp;publication_year=2004&amp;author=Greentree%2CAD\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"3.\">\n<p class=\"c-article-references__text\" id=\"ref-CR3\">Ralph, T. C., Resch, K. J. &amp; Gilchrist, A. Efficient Toffoli gates using qudits. Phys. Rev. A <b>75<\/b>, 022313 (2007).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevA.75.022313\" data-track-item_id=\"10.1103\/PhysRevA.75.022313\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevA.75.022313\" aria-label=\"Article reference 3\" data-doi=\"10.1103\/PhysRevA.75.022313\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2007PhRvA..75b2313R\" aria-label=\"ADS reference 3\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 3\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Efficient%20Toffoli%20gates%20using%20qudits&amp;journal=Phys.%20Rev.%20A&amp;doi=10.1103%2FPhysRevA.75.022313&amp;volume=75&amp;publication_year=2007&amp;author=Ralph%2CTC&amp;author=Resch%2CKJ&amp;author=Gilchrist%2CA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"4.\">\n<p class=\"c-article-references__text\" id=\"ref-CR4\">Fedorov, A., Steffen, L., Baur, M., Da Silva, M. P. &amp; Wallraff, A. Implementation of a Toffoli gate with superconducting circuits. Nature <b>481<\/b>, 170\u2013172 (2012).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/nature10713\" data-track-item_id=\"10.1038\/nature10713\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fnature10713\" aria-label=\"Article reference 4\" data-doi=\"10.1038\/nature10713\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2012Natur.481..170F\" aria-label=\"ADS reference 4\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BC3MXhs1aisLnJ\" aria-label=\"CAS reference 4\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 4\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Implementation%20of%20a%20Toffoli%20gate%20with%20superconducting%20circuits&amp;journal=Nature&amp;doi=10.1038%2Fnature10713&amp;volume=481&amp;pages=170-172&amp;publication_year=2012&amp;author=Fedorov%2CA&amp;author=Steffen%2CL&amp;author=Baur%2CM&amp;author=Silva%2CMP&amp;author=Wallraff%2CA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"5.\">\n<p class=\"c-article-references__text\" id=\"ref-CR5\">Bocharov, A., Roetteler, M. &amp; Svore, K. M. Factoring with qutrits: Shor\u2019s algorithm on ternary and metaplectic quantum architectures. Phys. Rev. A <b>96<\/b>, 012306 (2017).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevA.96.012306\" data-track-item_id=\"10.1103\/PhysRevA.96.012306\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevA.96.012306\" aria-label=\"Article reference 5\" data-doi=\"10.1103\/PhysRevA.96.012306\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2017PhRvA..96a2306B\" aria-label=\"ADS reference 5\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 5\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Factoring%20with%20qutrits%3A%20Shor%E2%80%99s%20algorithm%20on%20ternary%20and%20metaplectic%20quantum%20architectures&amp;journal=Phys.%20Rev.%20A&amp;doi=10.1103%2FPhysRevA.96.012306&amp;volume=96&amp;publication_year=2017&amp;author=Bocharov%2CA&amp;author=Roetteler%2CM&amp;author=Svore%2CKM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"6.\">\n<p class=\"c-article-references__text\" id=\"ref-CR6\">Gokhale, P. et al. Asymptotic improvements to quantum circuits via qutrits. In Proc. 46th International Symposium on Computer Architecture <a href=\"https:\/\/doi.org\/10.1145\/3307650.3322253\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.1145\/3307650.3322253\" target=\"_blank\" rel=\"noopener\">https:\/\/doi.org\/10.1145\/3307650.3322253<\/a> (ACM, 2019).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"7.\">\n<p class=\"c-article-references__text\" id=\"ref-CR7\">Chu, J. et al. Scalable algorithm simplification using quantum AND logic. Nat. Phys. <b>19<\/b>, 126\u2013131 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41567-022-01813-7\" data-track-item_id=\"10.1038\/s41567-022-01813-7\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41567-022-01813-7\" aria-label=\"Article reference 7\" data-doi=\"10.1038\/s41567-022-01813-7\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB38XivVOntLzO\" aria-label=\"CAS reference 7\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 7\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Scalable%20algorithm%20simplification%20using%20quantum%20AND%20logic&amp;journal=Nat.%20Phys.&amp;doi=10.1038%2Fs41567-022-01813-7&amp;volume=19&amp;pages=126-131&amp;publication_year=2023&amp;author=Chu%2CJ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"8.\">\n<p class=\"c-article-references__text\" id=\"ref-CR8\">Fern\u00e1ndez De Fuentes, I. et al. Navigating the 16-dimensional Hilbert space of a high-spin donor qudit with electric and magnetic fields. Nat. Commun. <b>15<\/b>, 1380 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41467-024-45368-y\" data-track-item_id=\"10.1038\/s41467-024-45368-y\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41467-024-45368-y\" aria-label=\"Article reference 8\" data-doi=\"10.1038\/s41467-024-45368-y\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024NatCo..15.1380F\" aria-label=\"ADS reference 8\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=38355747\" aria-label=\"PubMed reference 8\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC11258329\" aria-label=\"PubMed Central reference 8\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 8\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Navigating%20the%2016-dimensional%20Hilbert%20space%20of%20a%20high-spin%20donor%20qudit%20with%20electric%20and%20magnetic%20fields&amp;journal=Nat.%20Commun.&amp;doi=10.1038%2Fs41467-024-45368-y&amp;volume=15&amp;publication_year=2024&amp;author=Fern%C3%A1ndez%20De%20Fuentes%2CI\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"9.\">\n<p class=\"c-article-references__text\" id=\"ref-CR9\">Vilas, N. B. et al. An optical tweezer array of ultracold polyatomic molecules. Nature <b>628<\/b>, 282\u2013286 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41586-024-07199-1\" data-track-item_id=\"10.1038\/s41586-024-07199-1\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41586-024-07199-1\" aria-label=\"Article reference 9\" data-doi=\"10.1038\/s41586-024-07199-1\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024Natur.628..282V\" aria-label=\"ADS reference 9\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2cXnsVKktb4%3D\" aria-label=\"CAS reference 9\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=38570690\" aria-label=\"PubMed reference 9\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 9\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=An%20optical%20tweezer%20array%20of%20ultracold%20polyatomic%20molecules&amp;journal=Nature&amp;doi=10.1038%2Fs41586-024-07199-1&amp;volume=628&amp;pages=282-286&amp;publication_year=2024&amp;author=Vilas%2CNB\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"10.\">\n<p class=\"c-article-references__text\" id=\"ref-CR10\">Chaudhury, S. et al. Quantum control of the hyperfine spin of a Cs atom ensemble. Phys. Rev. Lett. <b>99<\/b>, 163002 (2007).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevLett.99.163002\" data-track-item_id=\"10.1103\/PhysRevLett.99.163002\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevLett.99.163002\" aria-label=\"Article reference 10\" data-doi=\"10.1103\/PhysRevLett.99.163002\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2007PhRvL..99p3002C\" aria-label=\"ADS reference 10\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=17995247\" aria-label=\"PubMed reference 10\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 10\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20control%20of%20the%20hyperfine%20spin%20of%20a%20Cs%20atom%20ensemble&amp;journal=Phys.%20Rev.%20Lett.&amp;doi=10.1103%2FPhysRevLett.99.163002&amp;volume=99&amp;publication_year=2007&amp;author=Chaudhury%2CS\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"11.\">\n<p class=\"c-article-references__text\" id=\"ref-CR11\">Kues, M. et al. On-chip generation of high-dimensional entangled quantum states and their coherent control. Nature <b>546<\/b>, 622\u2013626 (2017).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/nature22986\" data-track-item_id=\"10.1038\/nature22986\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fnature22986\" aria-label=\"Article reference 11\" data-doi=\"10.1038\/nature22986\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2017Natur.546..622K\" aria-label=\"ADS reference 11\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BC2sXhtVymsLvL\" aria-label=\"CAS reference 11\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=28658228\" aria-label=\"PubMed reference 11\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 11\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=On-chip%20generation%20of%20high-dimensional%20entangled%20quantum%20states%20and%20their%20coherent%20control&amp;journal=Nature&amp;doi=10.1038%2Fnature22986&amp;volume=546&amp;pages=622-626&amp;publication_year=2017&amp;author=Kues%2CM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"12.\">\n<p class=\"c-article-references__text\" id=\"ref-CR12\">Chi, Y. et al. A programmable qudit-based quantum processor. Nat. Commun. <b>13<\/b>, 1166 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41467-022-28767-x\" data-track-item_id=\"10.1038\/s41467-022-28767-x\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41467-022-28767-x\" aria-label=\"Article reference 12\" data-doi=\"10.1038\/s41467-022-28767-x\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022NatCo..13.1166C\" aria-label=\"ADS reference 12\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB38XmtVeqsr0%3D\" aria-label=\"CAS reference 12\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=35246519\" aria-label=\"PubMed reference 12\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC8897515\" aria-label=\"PubMed Central reference 12\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 12\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=A%20programmable%20qudit-based%20quantum%20processor&amp;journal=Nat.%20Commun.&amp;doi=10.1038%2Fs41467-022-28767-x&amp;volume=13&amp;publication_year=2022&amp;author=Chi%2CY\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"13.\">\n<p class=\"c-article-references__text\" id=\"ref-CR13\">Nguyen, L. B. et al. Empowering a qudit-based quantum processor by traversing the dual bosonic ladder. Nat. Commun. <b>15<\/b>, 7117 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41467-024-51434-2\" data-track-item_id=\"10.1038\/s41467-024-51434-2\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41467-024-51434-2\" aria-label=\"Article reference 13\" data-doi=\"10.1038\/s41467-024-51434-2\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2cXhvVensbrN\" aria-label=\"CAS reference 13\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=39160166\" aria-label=\"PubMed reference 13\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC11333499\" aria-label=\"PubMed Central reference 13\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 13\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Empowering%20a%20qudit-based%20quantum%20processor%20by%20traversing%20the%20dual%20bosonic%20ladder&amp;journal=Nat.%20Commun.&amp;doi=10.1038%2Fs41467-024-51434-2&amp;volume=15&amp;publication_year=2024&amp;author=Nguyen%2CLB\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"14.\">\n<p class=\"c-article-references__text\" id=\"ref-CR14\">Roy, S. et al. Synthetic high angular momentum spin dynamics in a microwave oscillator. Phys. Rev. X <b>15<\/b>, 021009 (2025).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"15.\">\n<p class=\"c-article-references__text\" id=\"ref-CR15\">Wang, Z., Parker, R. W., Champion, E. &amp; Blok, M. S. High-EJ\/EC transmon qudits with up to 12 levels. Phys. Rev. Appl. <b>23<\/b>, 034046 (2025).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"16.\">\n<p class=\"c-article-references__text\" id=\"ref-CR16\">Leupold, F. M. et al. Sustained state-independent quantum contextual correlations from a single ion. Phys. Rev. Lett. <b>120<\/b>, 180401 (2018).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevLett.120.180401\" data-track-item_id=\"10.1103\/PhysRevLett.120.180401\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevLett.120.180401\" aria-label=\"Article reference 16\" data-doi=\"10.1103\/PhysRevLett.120.180401\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2018PhRvL.120r0401L\" aria-label=\"ADS reference 16\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BC1MXltV2jsrg%3D\" aria-label=\"CAS reference 16\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=29775358\" aria-label=\"PubMed reference 16\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 16\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Sustained%20state-independent%20quantum%20contextual%20correlations%20from%20a%20single%20ion&amp;journal=Phys.%20Rev.%20Lett.&amp;doi=10.1103%2FPhysRevLett.120.180401&amp;volume=120&amp;publication_year=2018&amp;author=Leupold%2CFM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"17.\">\n<p class=\"c-article-references__text\" id=\"ref-CR17\">Ringbauer, M. et al. A universal qudit quantum processor with trapped ions. Nat. Phys. <b>18<\/b>, 1053\u20131057 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41567-022-01658-0\" data-track-item_id=\"10.1038\/s41567-022-01658-0\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41567-022-01658-0\" aria-label=\"Article reference 17\" data-doi=\"10.1038\/s41567-022-01658-0\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB38XhvV2lt7%2FE\" aria-label=\"CAS reference 17\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 17\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=A%20universal%20qudit%20quantum%20processor%20with%20trapped%20ions&amp;journal=Nat.%20Phys.&amp;doi=10.1038%2Fs41567-022-01658-0&amp;volume=18&amp;pages=1053-1057&amp;publication_year=2022&amp;author=Ringbauer%2CM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"18.\">\n<p class=\"c-article-references__text\" id=\"ref-CR18\">Adambukulam, C., Johnson, B., Morello, A. &amp; Laucht, A. Hyperfine spectroscopy and fast, all-optical arbitrary state initialization and readout of a single, ten-level 73Ge vacancy nuclear spin qudit in diamond. Phys. Rev. Lett. <b>132<\/b>, 060603 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevLett.132.060603\" data-track-item_id=\"10.1103\/PhysRevLett.132.060603\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevLett.132.060603\" aria-label=\"Article reference 18\" data-doi=\"10.1103\/PhysRevLett.132.060603\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024PhRvL.132f0603A\" aria-label=\"ADS reference 18\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2cXks1elsLs%3D\" aria-label=\"CAS reference 18\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=38394595\" aria-label=\"PubMed reference 18\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 18\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Hyperfine%20spectroscopy%20and%20fast%2C%20all-optical%20arbitrary%20state%20initialization%20and%20readout%20of%20a%20single%2C%20ten-level%2073Ge%20vacancy%20nuclear%20spin%20qudit%20in%20diamond&amp;journal=Phys.%20Rev.%20Lett.&amp;doi=10.1103%2FPhysRevLett.132.060603&amp;volume=132&amp;publication_year=2024&amp;author=Adambukulam%2CC&amp;author=Johnson%2CB&amp;author=Morello%2CA&amp;author=Laucht%2CA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"19.\">\n<p class=\"c-article-references__text\" id=\"ref-CR19\">Soltamov, V. A. et al. Excitation and coherent control of spin qudit modes in silicon carbide at room temperature. Nat. Commun. <b>10<\/b>, 1678 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41467-019-09429-x\" data-track-item_id=\"10.1038\/s41467-019-09429-x\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41467-019-09429-x\" aria-label=\"Article reference 19\" data-doi=\"10.1038\/s41467-019-09429-x\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2019NatCo..10.1678S\" aria-label=\"ADS reference 19\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:STN:280:DC%2BB3M%2FksFeluw%3D%3D\" aria-label=\"CAS reference 19\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=30975985\" aria-label=\"PubMed reference 19\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC6459825\" aria-label=\"PubMed Central reference 19\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 19\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Excitation%20and%20coherent%20control%20of%20spin%20qudit%20modes%20in%20silicon%20carbide%20at%20room%20temperature&amp;journal=Nat.%20Commun.&amp;doi=10.1038%2Fs41467-019-09429-x&amp;volume=10&amp;publication_year=2019&amp;author=Soltamov%2CVA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"20.\">\n<p class=\"c-article-references__text\" id=\"ref-CR20\">Gottesman, D., Kitaev, A. &amp; Preskill, J. Encoding a qubit in an oscillator. Phys. Rev. A <b>64<\/b>, 012310 (2001).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevA.64.012310\" data-track-item_id=\"10.1103\/PhysRevA.64.012310\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevA.64.012310\" aria-label=\"Article reference 20\" data-doi=\"10.1103\/PhysRevA.64.012310\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2001PhRvA..64a2310G\" aria-label=\"ADS reference 20\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 20\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Encoding%20a%20qubit%20in%20an%20oscillator&amp;journal=Phys.%20Rev.%20A&amp;doi=10.1103%2FPhysRevA.64.012310&amp;volume=64&amp;publication_year=2001&amp;author=Gottesman%2CD&amp;author=Kitaev%2CA&amp;author=Preskill%2CJ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"21.\">\n<p class=\"c-article-references__text\" id=\"ref-CR21\">Sivak, V. et al. Model-free quantum control with reinforcement learning. Phys. Rev. X <b>12<\/b>, 011059 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB38XpsFKkt7s%3D\" aria-label=\"CAS reference 21\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 21\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Model-free%20quantum%20control%20with%20reinforcement%20learning&amp;journal=Phys.%20Rev.%20X&amp;volume=12&amp;publication_year=2022&amp;author=Sivak%2CV\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"22.\">\n<p class=\"c-article-references__text\" id=\"ref-CR22\">Sivak, V. V. et al. Real-time quantum error correction beyond break-even. Nature <b>616<\/b>, 50\u201355 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41586-023-05782-6\" data-track-item_id=\"10.1038\/s41586-023-05782-6\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41586-023-05782-6\" aria-label=\"Article reference 22\" data-doi=\"10.1038\/s41586-023-05782-6\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023Natur.616...50S\" aria-label=\"ADS reference 22\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB3sXlvFGqs7o%3D\" aria-label=\"CAS reference 22\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=36949196\" aria-label=\"PubMed reference 22\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 22\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Real-time%20quantum%20error%20correction%20beyond%20break-even&amp;journal=Nature&amp;doi=10.1038%2Fs41586-023-05782-6&amp;volume=616&amp;pages=50-55&amp;publication_year=2023&amp;author=Sivak%2CVV\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"23.\">\n<p class=\"c-article-references__text\" id=\"ref-CR23\">Wang, Y., Hu, Z., Sanders, B. C. &amp; Kais, S. Qudits and high-dimensional quantum computing. Front. Phys. <b>8<\/b>, 589504 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.3389\/fphy.2020.589504\" data-track-item_id=\"10.3389\/fphy.2020.589504\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.3389%2Ffphy.2020.589504\" aria-label=\"Article reference 23\" data-doi=\"10.3389\/fphy.2020.589504\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 23\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Qudits%20and%20high-dimensional%20quantum%20computing&amp;journal=Front.%20Phys.&amp;doi=10.3389%2Ffphy.2020.589504&amp;volume=8&amp;publication_year=2020&amp;author=Wang%2CY&amp;author=Hu%2CZ&amp;author=Sanders%2CBC&amp;author=Kais%2CS\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"24.\">\n<p class=\"c-article-references__text\" id=\"ref-CR24\">Campbell, E. T., Anwar, H. &amp; Browne, D. E. Magic-state distillation in all prime dimensions using quantum Reed-Muller codes. Phys. Rev. X <b>2<\/b>, 041021 (2012).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BC3sXit1Cmu7Y%3D\" aria-label=\"CAS reference 24\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 24\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Magic-state%20distillation%20in%20all%20prime%20dimensions%20using%20quantum%20Reed-Muller%20codes&amp;journal=Phys.%20Rev.%20X&amp;volume=2&amp;publication_year=2012&amp;author=Campbell%2CET&amp;author=Anwar%2CH&amp;author=Browne%2CDE\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"25.\">\n<p class=\"c-article-references__text\" id=\"ref-CR25\">Campbell, E. T. Enhanced fault-tolerant quantum computing in d-level systems. Phys. Rev. Lett. <b>113<\/b>, 230501 (2014).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevLett.113.230501\" data-track-item_id=\"10.1103\/PhysRevLett.113.230501\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevLett.113.230501\" aria-label=\"Article reference 25\" data-doi=\"10.1103\/PhysRevLett.113.230501\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2014PhRvL.113w0501C\" aria-label=\"ADS reference 25\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BC2MXhvVKis7w%3D\" aria-label=\"CAS reference 25\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=25526106\" aria-label=\"PubMed reference 25\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 25\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Enhanced%20fault-tolerant%20quantum%20computing%20in%20d-level%20systems&amp;journal=Phys.%20Rev.%20Lett.&amp;doi=10.1103%2FPhysRevLett.113.230501&amp;volume=113&amp;publication_year=2014&amp;author=Campbell%2CET\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"26.\">\n<p class=\"c-article-references__text\" id=\"ref-CR26\">Meth, M. et al. Simulating two-dimensional lattice gauge theories on a qudit quantum computer. Nat. Phys. <b>21<\/b>, 570\u2013576 (2025).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"27.\">\n<p class=\"c-article-references__text\" id=\"ref-CR27\">Sawaya, N. P. D. et al. Resource-efficient digital quantum simulation of d-level systems for photonic, vibrational, and spin-s Hamiltonians. npj Quantum Inf. <b>6<\/b>, 49 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41534-020-0278-0\" data-track-item_id=\"10.1038\/s41534-020-0278-0\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41534-020-0278-0\" aria-label=\"Article reference 27\" data-doi=\"10.1038\/s41534-020-0278-0\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020npjQI...6...49S\" aria-label=\"ADS reference 27\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 27\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Resource-efficient%20digital%20quantum%20simulation%20of%20d-level%20systems%20for%20photonic%2C%20vibrational%2C%20and%20spin-s%20Hamiltonians&amp;journal=npj%20Quantum%20Inf.&amp;doi=10.1038%2Fs41534-020-0278-0&amp;volume=6&amp;publication_year=2020&amp;author=Sawaya%2CNPD\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"28.\">\n<p class=\"c-article-references__text\" id=\"ref-CR28\">Ofek, N. et al. Extending the lifetime of a quantum bit with error correction in superconducting circuits. Nature <b>536<\/b>, 441\u2013445 (2016).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/nature18949\" data-track-item_id=\"10.1038\/nature18949\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fnature18949\" aria-label=\"Article reference 28\" data-doi=\"10.1038\/nature18949\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2016Natur.536..441O\" aria-label=\"ADS reference 28\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BC28Xht1GrurnL\" aria-label=\"CAS reference 28\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=27437573\" aria-label=\"PubMed reference 28\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 28\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Extending%20the%20lifetime%20of%20a%20quantum%20bit%20with%20error%20correction%20in%20superconducting%20circuits&amp;journal=Nature&amp;doi=10.1038%2Fnature18949&amp;volume=536&amp;pages=441-445&amp;publication_year=2016&amp;author=Ofek%2CN\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"29.\">\n<p class=\"c-article-references__text\" id=\"ref-CR29\">Ni, Z. et al. Beating the break-even point with a discrete-variable-encoded logical qubit. Nature <b>616<\/b>, 56\u201360 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41586-023-05784-4\" data-track-item_id=\"10.1038\/s41586-023-05784-4\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41586-023-05784-4\" aria-label=\"Article reference 29\" data-doi=\"10.1038\/s41586-023-05784-4\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023Natur.616...56N\" aria-label=\"ADS reference 29\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB3sXlvFGqs78%3D\" aria-label=\"CAS reference 29\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=36949191\" aria-label=\"PubMed reference 29\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC10076216\" aria-label=\"PubMed Central reference 29\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 29\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Beating%20the%20break-even%20point%20with%20a%20discrete-variable-encoded%20logical%20qubit&amp;journal=Nature&amp;doi=10.1038%2Fs41586-023-05784-4&amp;volume=616&amp;pages=56-60&amp;publication_year=2023&amp;author=Ni%2CZ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"30.\">\n<p class=\"c-article-references__text\" id=\"ref-CR30\">Google Quantum AI and Collaborators. Quantum error correction below the surface code threshold. Nature <b>638<\/b>, 920\u2013926 (2025).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"31.\">\n<p class=\"c-article-references__text\" id=\"ref-CR31\">Fl\u00fchmann, C. et al. Encoding a qubit in a trapped-ion mechanical oscillator. Nature <b>566<\/b>, 513\u2013517 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41586-019-0960-6\" data-track-item_id=\"10.1038\/s41586-019-0960-6\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41586-019-0960-6\" aria-label=\"Article reference 31\" data-doi=\"10.1038\/s41586-019-0960-6\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2019Natur.566..513F\" aria-label=\"ADS reference 31\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=30814715\" aria-label=\"PubMed reference 31\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 31\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Encoding%20a%20qubit%20in%20a%20trapped-ion%20mechanical%20oscillator&amp;journal=Nature&amp;doi=10.1038%2Fs41586-019-0960-6&amp;volume=566&amp;pages=513-517&amp;publication_year=2019&amp;author=Fl%C3%BChmann%2CC\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"32.\">\n<p class=\"c-article-references__text\" id=\"ref-CR32\">De Neeve, B., Nguyen, T.-L., Behrle, T. &amp; Home, J. P. Error correction of a logical grid state qubit by dissipative pumping. Nat. Phys. <b>18<\/b>, 296\u2013300 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41567-021-01487-7\" data-track-item_id=\"10.1038\/s41567-021-01487-7\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41567-021-01487-7\" aria-label=\"Article reference 32\" data-doi=\"10.1038\/s41567-021-01487-7\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 32\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Error%20correction%20of%20a%20logical%20grid%20state%20qubit%20by%20dissipative%20pumping&amp;journal=Nat.%20Phys.&amp;doi=10.1038%2Fs41567-021-01487-7&amp;volume=18&amp;pages=296-300&amp;publication_year=2022&amp;author=Neeve%2CB&amp;author=Nguyen%2CT-L&amp;author=Behrle%2CT&amp;author=Home%2CJP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"33.\">\n<p class=\"c-article-references__text\" id=\"ref-CR33\">Campagne-Ibarcq, P. et al. Quantum error correction of a qubit encoded in grid states of an oscillator. Nature <b>584<\/b>, 368\u2013372 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41586-020-2603-3\" data-track-item_id=\"10.1038\/s41586-020-2603-3\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41586-020-2603-3\" aria-label=\"Article reference 33\" data-doi=\"10.1038\/s41586-020-2603-3\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB3cXhs1Cju7bL\" aria-label=\"CAS reference 33\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=32814889\" aria-label=\"PubMed reference 33\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 33\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20error%20correction%20of%20a%20qubit%20encoded%20in%20grid%20states%20of%20an%20oscillator&amp;journal=Nature&amp;doi=10.1038%2Fs41586-020-2603-3&amp;volume=584&amp;pages=368-372&amp;publication_year=2020&amp;author=Campagne-Ibarcq%2CP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"34.\">\n<p class=\"c-article-references__text\" id=\"ref-CR34\">Lachance-Quirion, D. et al. Autonomous quantum error correction of Gottesman\u2013Kitaev\u2013Preskill states. Phys. Rev. Lett. <b>132<\/b>, 150607 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevLett.132.150607\" data-track-item_id=\"10.1103\/PhysRevLett.132.150607\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevLett.132.150607\" aria-label=\"Article reference 34\" data-doi=\"10.1103\/PhysRevLett.132.150607\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024PhRvL.132o0607L\" aria-label=\"ADS reference 34\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2cXptVeitrc%3D\" aria-label=\"CAS reference 34\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=38682990\" aria-label=\"PubMed reference 34\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 34\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Autonomous%20quantum%20error%20correction%20of%20Gottesman%E2%80%93Kitaev%E2%80%93Preskill%20states&amp;journal=Phys.%20Rev.%20Lett.&amp;doi=10.1103%2FPhysRevLett.132.150607&amp;volume=132&amp;publication_year=2024&amp;author=Lachance-Quirion%2CD\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"35.\">\n<p class=\"c-article-references__text\" id=\"ref-CR35\">Konno, S. et al. Logical states for fault-tolerant quantum computation with propagating light. Science <b>383<\/b>, 289\u2013293 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1126\/science.adk7560\" data-track-item_id=\"10.1126\/science.adk7560\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1126%2Fscience.adk7560\" aria-label=\"Article reference 35\" data-doi=\"10.1126\/science.adk7560\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024Sci...383..289K\" aria-label=\"ADS reference 35\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4694976\" aria-label=\"MathSciNet reference 35\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2cXhvVCrsbw%3D\" aria-label=\"CAS reference 35\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=38236963\" aria-label=\"PubMed reference 35\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 35\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Logical%20states%20for%20fault-tolerant%20quantum%20computation%20with%20propagating%20light&amp;journal=Science&amp;doi=10.1126%2Fscience.adk7560&amp;volume=383&amp;pages=289-293&amp;publication_year=2024&amp;author=Konno%2CS\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"36.\">\n<p class=\"c-article-references__text\" id=\"ref-CR36\">Matsos, V. G. et al. Universal quantum gate set for Gottesman\u2013Kitaev\u2013Preskill logical qubits. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2409.05455\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2409.05455\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2409.05455<\/a> (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"37.\">\n<p class=\"c-article-references__text\" id=\"ref-CR37\">Gertler, J. M. et al. Protecting a bosonic qubit with autonomous quantum error correction. Nature <b>590<\/b>, 243\u2013248 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41586-021-03257-0\" data-track-item_id=\"10.1038\/s41586-021-03257-0\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41586-021-03257-0\" aria-label=\"Article reference 37\" data-doi=\"10.1038\/s41586-021-03257-0\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021Natur.590..243G\" aria-label=\"ADS reference 37\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB3MXjslKns7s%3D\" aria-label=\"CAS reference 37\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=33568826\" aria-label=\"PubMed reference 37\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 37\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Protecting%20a%20bosonic%20qubit%20with%20autonomous%20quantum%20error%20correction&amp;journal=Nature&amp;doi=10.1038%2Fs41586-021-03257-0&amp;volume=590&amp;pages=243-248&amp;publication_year=2021&amp;author=Gertler%2CJM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"38.\">\n<p class=\"c-article-references__text\" id=\"ref-CR38\">Place, A. P. M. et al. New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds. Nat. Commun. <b>12<\/b>, 1779 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41467-021-22030-5\" data-track-item_id=\"10.1038\/s41467-021-22030-5\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41467-021-22030-5\" aria-label=\"Article reference 38\" data-doi=\"10.1038\/s41467-021-22030-5\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021NatCo..12.1779P\" aria-label=\"ADS reference 38\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB3MXntl2rtb8%3D\" aria-label=\"CAS reference 38\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=33741989\" aria-label=\"PubMed reference 38\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC7979772\" aria-label=\"PubMed Central reference 38\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 38\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=New%20material%20platform%20for%20superconducting%20transmon%20qubits%20with%20coherence%20times%20exceeding%200.3%20milliseconds&amp;journal=Nat.%20Commun.&amp;doi=10.1038%2Fs41467-021-22030-5&amp;volume=12&amp;publication_year=2021&amp;author=Place%2CAPM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"39.\">\n<p class=\"c-article-references__text\" id=\"ref-CR39\">Ganjam, S. et al. Surpassing millisecond coherence in on chip superconducting quantum memories by optimizing materials and circuit design. Nat. Commun. <b>15<\/b>, 3687 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41467-024-47857-6\" data-track-item_id=\"10.1038\/s41467-024-47857-6\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41467-024-47857-6\" aria-label=\"Article reference 39\" data-doi=\"10.1038\/s41467-024-47857-6\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024NatCo..15.3687G\" aria-label=\"ADS reference 39\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2cXpvFKks7k%3D\" aria-label=\"CAS reference 39\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=38693124\" aria-label=\"PubMed reference 39\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC11063213\" aria-label=\"PubMed Central reference 39\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 39\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Surpassing%20millisecond%20coherence%20in%20on%20chip%20superconducting%20quantum%20memories%20by%20optimizing%20materials%20and%20circuit%20design&amp;journal=Nat.%20Commun.&amp;doi=10.1038%2Fs41467-024-47857-6&amp;volume=15&amp;publication_year=2024&amp;author=Ganjam%2CS\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"40.\">\n<p class=\"c-article-references__text\" id=\"ref-CR40\">Reagor, M. et al. Quantum memory with millisecond coherence in circuit QED. Phys. Rev. B <b>94<\/b>, 014506 (2016).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevB.94.014506\" data-track-item_id=\"10.1103\/PhysRevB.94.014506\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevB.94.014506\" aria-label=\"Article reference 40\" data-doi=\"10.1103\/PhysRevB.94.014506\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2016PhRvB..94a4506R\" aria-label=\"ADS reference 40\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 40\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20memory%20with%20millisecond%20coherence%20in%20circuit%20QED&amp;journal=Phys.%20Rev.%20B&amp;doi=10.1103%2FPhysRevB.94.014506&amp;volume=94&amp;publication_year=2016&amp;author=Reagor%2CM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"41.\">\n<p class=\"c-article-references__text\" id=\"ref-CR41\">Weyl, H. The Theory of Groups and Quantum Mechanics (Dover Publications, 1950).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"42.\">\n<p class=\"c-article-references__text\" id=\"ref-CR42\">Schwinger, J. Unitary operator bases. Proc. Natl Acad. Sci. USA <b>46<\/b>, 570 (1960).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1073\/pnas.46.4.570\" data-track-item_id=\"10.1073\/pnas.46.4.570\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1073%2Fpnas.46.4.570\" aria-label=\"Article reference 42\" data-doi=\"10.1073\/pnas.46.4.570\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=1960PNAS...46..570S\" aria-label=\"ADS reference 42\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=115648\" aria-label=\"MathSciNet reference 42\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:STN:280:DC%2BD28zitF2juw%3D%3D\" aria-label=\"CAS reference 42\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=16590645\" aria-label=\"PubMed reference 42\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC222876\" aria-label=\"PubMed Central reference 42\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 42\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Unitary%20operator%20bases&amp;journal=Proc.%20Natl%20Acad.%20Sci.%20USA&amp;doi=10.1073%2Fpnas.46.4.570&amp;volume=46&amp;publication_year=1960&amp;author=Schwinger%2CJ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"43.\">\n<p class=\"c-article-references__text\" id=\"ref-CR43\">Royer, B., Singh, S. &amp; Girvin, S. Stabilization of finite-energy Gottesman-Kitaev-Preskill states. Phys. Rev. Lett. <b>125<\/b>, 260509 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevLett.125.260509\" data-track-item_id=\"10.1103\/PhysRevLett.125.260509\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevLett.125.260509\" aria-label=\"Article reference 43\" data-doi=\"10.1103\/PhysRevLett.125.260509\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020PhRvL.125z0509R\" aria-label=\"ADS reference 43\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB3MXivVOkt7o%3D\" aria-label=\"CAS reference 43\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=33449723\" aria-label=\"PubMed reference 43\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 43\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Stabilization%20of%20finite-energy%20Gottesman-Kitaev-Preskill%20states&amp;journal=Phys.%20Rev.%20Lett.&amp;doi=10.1103%2FPhysRevLett.125.260509&amp;volume=125&amp;publication_year=2020&amp;author=Royer%2CB&amp;author=Singh%2CS&amp;author=Girvin%2CS\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"44.\">\n<p class=\"c-article-references__text\" id=\"ref-CR44\">Grimsmo, A. L. &amp; Puri, S. Quantum error correction with the Gottesman-Kitaev-Preskill code. PRX Quantum <b>2<\/b>, 020101 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PRXQuantum.2.020101\" data-track-item_id=\"10.1103\/PRXQuantum.2.020101\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPRXQuantum.2.020101\" aria-label=\"Article reference 44\" data-doi=\"10.1103\/PRXQuantum.2.020101\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021PRXQ....2b0101G\" aria-label=\"ADS reference 44\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 44\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20error%20correction%20with%20the%20Gottesman-Kitaev-Preskill%20code&amp;journal=PRX%20Quantum&amp;doi=10.1103%2FPRXQuantum.2.020101&amp;volume=2&amp;publication_year=2021&amp;author=Grimsmo%2CAL&amp;author=Puri%2CS\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"45.\">\n<p class=\"c-article-references__text\" id=\"ref-CR45\">Fl\u00fchmann, C. &amp; Home, J. Direct characteristic-function tomography of quantum states of the trapped-ion motional oscillator. Phys. Rev. Lett. <b>125<\/b>, 043602 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevLett.125.043602\" data-track-item_id=\"10.1103\/PhysRevLett.125.043602\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevLett.125.043602\" aria-label=\"Article reference 45\" data-doi=\"10.1103\/PhysRevLett.125.043602\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020PhRvL.125d3602F\" aria-label=\"ADS reference 45\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=32794777\" aria-label=\"PubMed reference 45\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 45\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Direct%20characteristic-function%20tomography%20of%20quantum%20states%20of%20the%20trapped-ion%20motional%20oscillator&amp;journal=Phys.%20Rev.%20Lett.&amp;doi=10.1103%2FPhysRevLett.125.043602&amp;volume=125&amp;publication_year=2020&amp;author=Fl%C3%BChmann%2CC&amp;author=Home%2CJ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"46.\">\n<p class=\"c-article-references__text\" id=\"ref-CR46\">Kenfack, A. &amp; \u017byczkowski, K. Negativity of the Wigner function as an indicator of non-classicality. J. Opt. B\u00a0<b>6<\/b>, 396 (2004).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1088\/1464-4266\/6\/10\/003\" data-track-item_id=\"10.1088\/1464-4266\/6\/10\/003\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1088%2F1464-4266%2F6%2F10%2F003\" aria-label=\"Article reference 46\" data-doi=\"10.1088\/1464-4266\/6\/10\/003\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2004JOptB...6..396K\" aria-label=\"ADS reference 46\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=2097755\" aria-label=\"MathSciNet reference 46\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 46\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Negativity%20of%20the%20Wigner%20function%20as%20an%20indicator%20of%20non-classicality&amp;journal=J.%20Opt.%20B&amp;doi=10.1088%2F1464-4266%2F6%2F10%2F003&amp;volume=6&amp;publication_year=2004&amp;author=Kenfack%2CA&amp;author=%C5%BByczkowski%2CK\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"47.\">\n<p class=\"c-article-references__text\" id=\"ref-CR47\">Nielsen, M. A. A simple formula for the average gate fidelity of a quantum dynamical operation. Phys. Lett. A <b>303<\/b>, 249\u2013252 (2002).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1016\/S0375-9601(02)01272-0\" data-track-item_id=\"10.1016\/S0375-9601(02)01272-0\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1016%2FS0375-9601%2802%2901272-0\" aria-label=\"Article reference 47\" data-doi=\"10.1016\/S0375-9601(02)01272-0\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2002PhLA..303..249N\" aria-label=\"ADS reference 47\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=1937100\" aria-label=\"MathSciNet reference 47\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BD38XnvFylsbo%3D\" aria-label=\"CAS reference 47\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 47\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=A%20simple%20formula%20for%20the%20average%20gate%20fidelity%20of%20a%20quantum%20dynamical%20operation&amp;journal=Phys.%20Lett.%20A&amp;doi=10.1016%2FS0375-9601%2802%2901272-0&amp;volume=303&amp;pages=249-252&amp;publication_year=2002&amp;author=Nielsen%2CMA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"48.\">\n<p class=\"c-article-references__text\" id=\"ref-CR48\">Eickbusch, A. et al. Fast universal control of an oscillator with weak dispersive coupling to a qubit. Nat. Phys. <b>18<\/b>, 1464\u20131469 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41567-022-01776-9\" data-track-item_id=\"10.1038\/s41567-022-01776-9\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41567-022-01776-9\" aria-label=\"Article reference 48\" data-doi=\"10.1038\/s41567-022-01776-9\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB38Xis1CrurjL\" aria-label=\"CAS reference 48\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 48\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Fast%20universal%20control%20of%20an%20oscillator%20with%20weak%20dispersive%20coupling%20to%20a%20qubit&amp;journal=Nat.%20Phys.&amp;doi=10.1038%2Fs41567-022-01776-9&amp;volume=18&amp;pages=1464-1469&amp;publication_year=2022&amp;author=Eickbusch%2CA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"49.\">\n<p class=\"c-article-references__text\" id=\"ref-CR49\">Rosenblum, S. et al. Fault-tolerant detection of a quantum error. Science <b>361<\/b>, 266\u2013270 (2018).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1126\/science.aat3996\" data-track-item_id=\"10.1126\/science.aat3996\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1126%2Fscience.aat3996\" aria-label=\"Article reference 49\" data-doi=\"10.1126\/science.aat3996\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2018Sci...361..266R\" aria-label=\"ADS reference 49\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=3821892\" aria-label=\"MathSciNet reference 49\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BC1cXhtlantbvO\" aria-label=\"CAS reference 49\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=30026224\" aria-label=\"PubMed reference 49\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 49\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Fault-tolerant%20detection%20of%20a%20quantum%20error&amp;journal=Science&amp;doi=10.1126%2Fscience.aat3996&amp;volume=361&amp;pages=266-270&amp;publication_year=2018&amp;author=Rosenblum%2CS\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"50.\">\n<p class=\"c-article-references__text\" id=\"ref-CR50\">Ding, A. Z. et al. Quantum control of an oscillator with a Kerr-cat qubit. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2407.10940\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2407.10940\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2407.10940<\/a> (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"51.\">\n<p class=\"c-article-references__text\" id=\"ref-CR51\">Schmidt, F., Miller, D. &amp; Van Loock, P. Error-corrected quantum repeaters with Gottesman\u2013Kitaev\u2013Preskill qudits. Phys. Rev. A <b>109<\/b>, 042427 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevA.109.042427\" data-track-item_id=\"10.1103\/PhysRevA.109.042427\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevA.109.042427\" aria-label=\"Article reference 51\" data-doi=\"10.1103\/PhysRevA.109.042427\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024PhRvA.109d2427S\" aria-label=\"ADS reference 51\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4740577\" aria-label=\"MathSciNet reference 51\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2cXhtVansrrF\" aria-label=\"CAS reference 51\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 51\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Error-corrected%20quantum%20repeaters%20with%20Gottesman%E2%80%93Kitaev%E2%80%93Preskill%20qudits&amp;journal=Phys.%20Rev.%20A&amp;doi=10.1103%2FPhysRevA.109.042427&amp;volume=109&amp;publication_year=2024&amp;author=Schmidt%2CF&amp;author=Miller%2CD&amp;author=Loock%2CP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"52.\">\n<p class=\"c-article-references__text\" id=\"ref-CR52\">Wang, Z. &amp; Jiang, L. Passive environment-assisted quantum communication with GKP states. Phys. Rev. X <b>15<\/b>, 021003 (2025).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"53.\">\n<p class=\"c-article-references__text\" id=\"ref-CR53\">Terhal, B. M., Conrad, J. &amp; Vuillot, C. Towards scalable bosonic quantum error correction. Quantum Sci. Technol. <b>5<\/b>, 043001 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1088\/2058-9565\/ab98a5\" data-track-item_id=\"10.1088\/2058-9565\/ab98a5\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1088%2F2058-9565%2Fab98a5\" aria-label=\"Article reference 53\" data-doi=\"10.1088\/2058-9565\/ab98a5\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020QS%26T....5d3001T\" aria-label=\"ADS reference 53\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 53\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Towards%20scalable%20bosonic%20quantum%20error%20correction&amp;journal=Quantum%20Sci.%20Technol.&amp;doi=10.1088%2F2058-9565%2Fab98a5&amp;volume=5&amp;publication_year=2020&amp;author=Terhal%2CBM&amp;author=Conrad%2CJ&amp;author=Vuillot%2CC\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"54.\">\n<p class=\"c-article-references__text\" id=\"ref-CR54\">Schmidt, F. &amp; van Loock, P. Quantum error correction with higher Gottesman\u2013Kitaev\u2013Preskill codes: minimal measurements and linear optics. Phys. Rev. A <b>105<\/b>, 042427 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevA.105.042427\" data-track-item_id=\"10.1103\/PhysRevA.105.042427\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevA.105.042427\" aria-label=\"Article reference 54\" data-doi=\"10.1103\/PhysRevA.105.042427\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022PhRvA.105d2427S\" aria-label=\"ADS reference 54\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4420967\" aria-label=\"MathSciNet reference 54\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB38Xhtlagtb3N\" aria-label=\"CAS reference 54\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 54\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20error%20correction%20with%20higher%20Gottesman%E2%80%93Kitaev%E2%80%93Preskill%20codes%3A%20minimal%20measurements%20and%20linear%20optics&amp;journal=Phys.%20Rev.%20A&amp;doi=10.1103%2FPhysRevA.105.042427&amp;volume=105&amp;publication_year=2022&amp;author=Schmidt%2CF&amp;author=Loock%2CP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"55.\">\n<p class=\"c-article-references__text\" id=\"ref-CR55\">Rojkov, I. et al. Two-qubit operations for finite-energy Gottesman\u2013Kitaev\u2013Preskill encodings. Phys. Rev. Lett. <b>133<\/b>, 100601 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevLett.133.100601\" data-track-item_id=\"10.1103\/PhysRevLett.133.100601\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevLett.133.100601\" aria-label=\"Article reference 55\" data-doi=\"10.1103\/PhysRevLett.133.100601\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=39303237\" aria-label=\"PubMed reference 55\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 55\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Two-qubit%20operations%20for%20finite-energy%20Gottesman%E2%80%93Kitaev%E2%80%93Preskill%20encodings&amp;journal=Phys.%20Rev.%20Lett.&amp;doi=10.1103%2FPhysRevLett.133.100601&amp;volume=133&amp;publication_year=2024&amp;author=Rojkov%2CI\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"56.\">\n<p class=\"c-article-references__text\" id=\"ref-CR56\">Cafaro, C., Maiolini, F. &amp; Mancini, S. Quantum stabilizer codes embedding qubits into qudits. Phys. Rev. A <b>86<\/b>, 022308 (2012).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevA.86.022308\" data-track-item_id=\"10.1103\/PhysRevA.86.022308\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevA.86.022308\" aria-label=\"Article reference 56\" data-doi=\"10.1103\/PhysRevA.86.022308\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2012PhRvA..86b2308C\" aria-label=\"ADS reference 56\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 56\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20stabilizer%20codes%20embedding%20qubits%20into%20qudits&amp;journal=Phys.%20Rev.%20A&amp;doi=10.1103%2FPhysRevA.86.022308&amp;volume=86&amp;publication_year=2012&amp;author=Cafaro%2CC&amp;author=Maiolini%2CF&amp;author=Mancini%2CS\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"57.\">\n<p class=\"c-article-references__text\" id=\"ref-CR57\">Gross, J. A. Designing codes around interactions: the case of a spin. Phys. Rev. Lett. <b>127<\/b>, 010504 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevLett.127.010504\" data-track-item_id=\"10.1103\/PhysRevLett.127.010504\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevLett.127.010504\" aria-label=\"Article reference 57\" data-doi=\"10.1103\/PhysRevLett.127.010504\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021PhRvL.127a0504G\" aria-label=\"ADS reference 57\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4289941\" aria-label=\"MathSciNet reference 57\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB3MXhs1WisrbI\" aria-label=\"CAS reference 57\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=34270305\" aria-label=\"PubMed reference 57\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 57\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Designing%20codes%20around%20interactions%3A%20the%20case%20of%20a%20spin&amp;journal=Phys.%20Rev.%20Lett.&amp;doi=10.1103%2FPhysRevLett.127.010504&amp;volume=127&amp;publication_year=2021&amp;author=Gross%2CJA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"58.\">\n<p class=\"c-article-references__text\" id=\"ref-CR58\">Gross, J. A., Godfrin, C., Blais, A. &amp; Dupont-Ferrier, E. Hardware-efficient error-correcting codes for large nuclear spins. Phys. Rev. Appl. <b>22<\/b>, 014006 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevApplied.22.014006\" data-track-item_id=\"10.1103\/PhysRevApplied.22.014006\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevApplied.22.014006\" aria-label=\"Article reference 58\" data-doi=\"10.1103\/PhysRevApplied.22.014006\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2cXit1Glu7fP\" aria-label=\"CAS reference 58\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 58\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Hardware-efficient%20error-correcting%20codes%20for%20large%20nuclear%20spins&amp;journal=Phys.%20Rev.%20Appl.&amp;doi=10.1103%2FPhysRevApplied.22.014006&amp;volume=22&amp;publication_year=2024&amp;author=Gross%2CJA&amp;author=Godfrin%2CC&amp;author=Blais%2CA&amp;author=Dupont-Ferrier%2CE\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"59.\">\n<p class=\"c-article-references__text\" id=\"ref-CR59\">Brock, B. et al. Data for \u2018Quantum error correction of qudits beyond break-even\u2019. Zenodo <a href=\"https:\/\/doi.org\/10.5281\/zenodo.15009817\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.5281\/zenodo.15009817\" target=\"_blank\" rel=\"noopener\">https:\/\/doi.org\/10.5281\/zenodo.15009817<\/a> (2025).<\/p>\n<\/li>\n","protected":false},"excerpt":{"rendered":"Blume-Kohout, R., Caves, C. M. &amp; Deutsch, I. H. Climbing mount scalable: physical resource requirements for a scalable&hellip;\n","protected":false},"author":2,"featured_media":101724,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3845],"tags":[3965,3966,74,7030,17844,70,16,15],"class_list":{"0":"post-101723","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-physics","8":"tag-humanities-and-social-sciences","9":"tag-multidisciplinary","10":"tag-physics","11":"tag-quantum-information","12":"tag-quantum-mechanics","13":"tag-science","14":"tag-uk","15":"tag-united-kingdom"},"share_on_mastodon":{"url":"https:\/\/pubeurope.com\/@uk\/114508136696147892","error":""},"_links":{"self":[{"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/posts\/101723","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/comments?post=101723"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/posts\/101723\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/media\/101724"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/media?parent=101723"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/categories?post=101723"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/tags?post=101723"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}