{"id":304391,"date":"2025-07-30T16:52:12","date_gmt":"2025-07-30T16:52:12","guid":{"rendered":"https:\/\/www.europesays.com\/uk\/304391\/"},"modified":"2025-07-30T16:52:12","modified_gmt":"2025-07-30T16:52:12","slug":"tensor-networks-for-quantum-computing","status":"publish","type":"post","link":"https:\/\/www.europesays.com\/uk\/304391\/","title":{"rendered":"Tensor networks for quantum computing"},"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\">Schollw\u00f6ck, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys. <b>326<\/b>, 96\u2013192 (2011).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2011AnPhy.326...96S\" aria-label=\"ADS reference 1\" 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=2769670\" 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=The%20density-matrix%20renormalization%20group%20in%20the%20age%20of%20matrix%20product%20states&amp;journal=Ann.%20Phys.&amp;volume=326&amp;pages=96-192&amp;publication_year=2011&amp;author=Schollw%C3%B6ck%2CU\" 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\">Or\u00fas, R. Tensor networks for complex quantum systems. Nat. Rev. Phys. <b>1<\/b>, 538\u2013550 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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=Tensor%20networks%20for%20complex%20quantum%20systems&amp;journal=Nat.%20Rev.%20Phys.&amp;volume=1&amp;pages=538-550&amp;publication_year=2019&amp;author=Or%C3%BAs%2CR\" 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\">Or\u00fas, R. A practical introduction to tensor networks: matrix product states and projected entangled pair states. Ann. Phys. <b>349<\/b>, 117\u2013158 (2014).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2014AnPhy.349..117O\" aria-label=\"ADS reference 3\" 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=3244614\" aria-label=\"MathSciNet reference 3\" 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 3\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=A%20practical%20introduction%20to%20tensor%20networks%3A%20matrix%20product%20states%20and%20projected%20entangled%20pair%20states&amp;journal=Ann.%20Phys.&amp;volume=349&amp;pages=117-158&amp;publication_year=2014&amp;author=Or%C3%BAs%2CR\" 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\">Chubb, C. T. General tensor network decoding of 2D Pauli codes. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2101.04125\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2101.04125\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2101.04125<\/a> (2021).<\/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\">Pan, F. &amp; Zhang, P. Simulation of quantum circuits using the big-batch tensor network method. Phys. Rev. Lett. <b>128<\/b>, 030501 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022PhRvL.128c0501P\" 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=Simulation%20of%20quantum%20circuits%20using%20the%20big-batch%20tensor%20network%20method&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=128&amp;publication_year=2022&amp;author=Pan%2CF&amp;author=Zhang%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=\"6.\">\n<p class=\"c-article-references__text\" id=\"ref-CR6\">Fu, R. et al. Achieving energetic superiority through system-level quantum circuit simulation. In SC &#8217;24: Proceedings of the International Conference for High Performance Computing, Networking, Storage, and Analysis 1\u201320 (ACM, 2024).<\/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\">Oh, C., Liu, M., Alexeev, Y., Fefferman, B. &amp; Jiang, L. Classical algorithm for simulating experimental Gaussian boson sampling. Nat. Phys. <b>20<\/b>, 1461\u20131468 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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=Classical%20algorithm%20for%20simulating%20experimental%20Gaussian%20boson%20sampling.&amp;journal=Nat.%20Phys.&amp;volume=20&amp;pages=1461-1468&amp;publication_year=2024&amp;author=Oh%2CC&amp;author=Liu%2CM&amp;author=Alexeev%2CY&amp;author=Fefferman%2CB&amp;author=Jiang%2CL\" 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\">Tindall, J., Fishman, M., Stoudenmire, E. M. &amp; Sels, D. Efficient tensor network simulation of IBM\u2019s Eagle kicked Ising experiment. PRX Quantum <b>5<\/b>, 010308 (2024).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024PRXQ....5a0308T\" aria-label=\"ADS reference 8\" 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 8\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Efficient%20tensor%20network%20simulation%20of%20IBM%E2%80%99s%20Eagle%20kicked%20Ising%20experiment&amp;journal=PRX%20Quantum&amp;volume=5&amp;publication_year=2024&amp;author=Tindall%2CJ&amp;author=Fishman%2CM&amp;author=Stoudenmire%2CEM&amp;author=Sels%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=\"9.\">\n<p class=\"c-article-references__text\" id=\"ref-CR9\">Tindall, J., Mello, A. F., Fishman, M., Stoudenmire, E. M. &amp; Sels, D. Dynamics of disordered quantum systems with two- and three-dimensional tensor networks. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2503.05693\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2503.05693\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2503.05693<\/a> (2025).<\/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\">Bridgeman, J. C. &amp; Chubb, C. T. Hand-waving and interpretive dance: an introductory course on tensor networks. J. Phys. A <b>50<\/b>, 223001 (2017). <b>A comprehensive and pedagogical introduction to tensor networks from the traditional many-body physics perspective.<\/b><\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2017JPhA...50v3001B\" 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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=3659106\" aria-label=\"MathSciNet reference 10\" 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 10\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Hand-waving%20and%20interpretive%20dance%3A%20an%20introductory%20course%20on%20tensor%20networks&amp;journal=J.%20Phys.%20A&amp;volume=50&amp;publication_year=2017&amp;author=Bridgeman%2CJC&amp;author=Chubb%2CCT\" 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\">Eckart, C. &amp; Young, G. The approximation of one matrix by another of lower rank. Psychometrika <b>1<\/b>, 211\u2013218 (1936).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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=The%20approximation%20of%20one%20matrix%20by%20another%20of%20lower%20rank&amp;journal=Psychometrika&amp;volume=1&amp;pages=211-218&amp;publication_year=1936&amp;author=Eckart%2CC&amp;author=Young%2CG\" 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\">Stewart, G. W. Matrix Algorithms, Vol. 1 Basic Decompositions (SIAM, 1998).<\/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\">Verstraete, F. &amp; Cirac, J. I. Matrix product states represent ground states faithfully. Phys. Rev. B <b>73<\/b>, 094423 (2006). <b>Analytical results showing that MPS efficiently represents the ground states of 1D systems.<\/b><\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2006PhRvB..73i4423V\" aria-label=\"ADS reference 13\" 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 13\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Matrix%20product%20states%20represent%20ground%20states%20faithfully&amp;journal=Phys.%20Rev.%20B&amp;volume=73&amp;publication_year=2006&amp;author=Verstraete%2CF&amp;author=Cirac%2CJI\" 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\">Ba\u00f1uls, M. C. Tensor network algorithms: a route map. Annu. Rev. Condens. Matter Phys. <b>14<\/b>, 173\u2013191 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023ARCMP..14..173B\" aria-label=\"ADS reference 14\" 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=4595687\" aria-label=\"MathSciNet reference 14\" 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 14\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tensor%20network%20algorithms%3A%20a%20route%20map&amp;journal=Annu.%20Rev.%20Condens.%20Matter%20Phys.&amp;volume=14&amp;pages=173-191&amp;publication_year=2023&amp;author=Ba%C3%B1uls%2CMC\" 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=\"15.\">\n<p class=\"c-article-references__text\" id=\"ref-CR15\">Oseledets, I. V. Tensor-train decomposition. SIAM J. Sci. Comput. <b>33<\/b>, 2295\u20132317 (2011).<\/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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=2837533\" aria-label=\"MathSciNet reference 15\" 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 15\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tensor-train%20decomposition&amp;journal=SIAM%20J.%20Sci.%20Comput.&amp;volume=33&amp;pages=2295-2317&amp;publication_year=2011&amp;author=Oseledets%2CIV\" 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=\"16.\">\n<p class=\"c-article-references__text\" id=\"ref-CR16\">Schuch, N., Wolf, M. M., Verstraete, F. &amp; Cirac, J. I. Computational complexity of projected entangled pair states. Phys. Rev. Lett. <b>98<\/b>, 140506 (2007). <b>Shows that contracting PEPS and arbitrary tensor networks is #P-complete.<\/b><\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2007PhRvL..98n0506S\" 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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=2308841\" aria-label=\"MathSciNet reference 16\" 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 16\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Computational%20complexity%20of%20projected%20entangled%20pair%20states&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=98&amp;publication_year=2007&amp;author=Schuch%2CN&amp;author=Wolf%2CMM&amp;author=Verstraete%2CF&amp;author=Cirac%2CJI\" 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\">Vidal, G. Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. <b>91<\/b>, 147902 (2003).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2003PhRvL..91n7902V\" aria-label=\"ADS reference 17\" 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 17\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Efficient%20classical%20simulation%20of%20slightly%20entangled%20quantum%20computations&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=91&amp;publication_year=2003&amp;author=Vidal%2CG\" 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\">Shi, Y.-Y., Duan, L.-M. &amp; Vidal, G. Classical simulation of quantum many-body systems with a tree tensor network. Phys. Rev. A <b>74<\/b>, 022320 (2006).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2006PhRvA..74b2320S\" aria-label=\"ADS reference 18\" 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 18\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Classical%20simulation%20of%20quantum%20many-body%20systems%20with%20a%20tree%20tensor%20network&amp;journal=Phys.%20Rev.%20A&amp;volume=74&amp;publication_year=2006&amp;author=Shi%2CY-Y&amp;author=Duan%2CL-M&amp;author=Vidal%2CG\" 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\">Wang, H. &amp; Thoss, M. Multilayer formulation of the multiconfiguration time-dependent Hartree theory. J. Chem. Phys. <b>119<\/b>, 1289\u20131299 (2003).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2003JChPh.119.1289W\" aria-label=\"ADS reference 19\" 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 19\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Multilayer%20formulation%20of%20the%20multiconfiguration%20time-dependent%20Hartree%20theory&amp;journal=J.%20Chem.%20Phys.&amp;volume=119&amp;pages=1289-1299&amp;publication_year=2003&amp;author=Wang%2CH&amp;author=Thoss%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=\"20.\">\n<p class=\"c-article-references__text\" id=\"ref-CR20\">Markov, I. L. &amp; Shi, Y. Simulating quantum computation by contracting tensor networks. SIAM J. Comput. <b>38<\/b>, 963\u2013981 (2008). <b>Proposes exact simulation of quantum circuits with tensor networks and links the complexity to the treewidth of the graph underlying the circuit.<\/b><\/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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=2421074\" aria-label=\"MathSciNet reference 20\" 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 20\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Simulating%20quantum%20computation%20by%20contracting%20tensor%20networks&amp;journal=SIAM%20J.%20Comput.&amp;volume=38&amp;pages=963-981&amp;publication_year=2008&amp;author=Markov%2CIL&amp;author=Shi%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=\"21.\">\n<p class=\"c-article-references__text\" id=\"ref-CR21\">DeCross, M. et al. The computational power of random quantum circuits in arbitrary geometries. Phys. Rev. X. <b>15<\/b>, 021052 (2025).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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=The%20computational%20power%20of%20random%20quantum%20circuits%20in%20arbitrary%20geometries&amp;journal=Phys.%20Rev.%20X.&amp;volume=15&amp;publication_year=2025&amp;author=DeCross%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=\"22.\">\n<p class=\"c-article-references__text\" id=\"ref-CR22\">Bayraktar, H. et al. cuQuantum SDK: a high-performance library for accelerating quantum science. Preprint at <a href=\"https:\/\/arxiv.org\/pdf\/2308.01999\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/pdf\/2308.01999\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/pdf\/2308.01999<\/a> (2023).<\/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\">Gray, J. quimb: A python package for quantum information and many-body calculations. J. Open Source Softw.\u00a0<b>3<\/b>, 819 (2018).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2018JOSS....3..819G\" aria-label=\"ADS reference 23\" 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 23\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=quimb%3A%20A%20python%20package%20for%20quantum%20information%20and%20many-body%20calculations&amp;journal=J.%20Open%20Source%20Softw.&amp;volume=3&amp;publication_year=2018&amp;author=Gray%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=\"24.\">\n<p class=\"c-article-references__text\" id=\"ref-CR24\">Fishman, M., White, S. &amp; Stoudenmire, E. The ITensor software library for tensor network calculations. SciPost Physics Codebases 004 (2022).<\/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\">Gray, J. &amp; Kourtis, S. Hyper-optimized tensor network contraction. Quantum <b>5<\/b>, 410 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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=Hyper-optimized%20tensor%20network%20contraction&amp;journal=Quantum&amp;volume=5&amp;publication_year=2021&amp;author=Gray%2CJ&amp;author=Kourtis%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=\"26.\">\n<p class=\"c-article-references__text\" id=\"ref-CR26\">Hauschild, J. &amp; Pollmann, F. Efficient numerical simulations with tensor networks: Tensor Network Python (TeNPy). SciPost Physics Lecture Notes 005 (2018).<\/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\">Zhai, H. et al. Block2: A comprehensive open source framework to develop and apply state-of-the-art dmrg algorithms in electronic structure and beyond. The Journal of Chemical Physics <b>159<\/b>, 234801 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023JChPh.159w4801Z\" 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=Block2%3A%20A%20comprehensive%20open%20source%20framework%20to%20develop%20and%20apply%20state-of-the-art%20dmrg%20algorithms%20in%20electronic%20structure%20and%20beyond&amp;journal=The%20Journal%20of%20Chemical%20Physics&amp;volume=159&amp;publication_year=2023&amp;author=Zhai%2CH\" 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\">White, S. R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. <b>69<\/b>, 2863 (1992).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=1992PhRvL..69.2863W\" aria-label=\"ADS reference 28\" 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 28\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Density%20matrix%20formulation%20for%20quantum%20renormalization%20groups&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=69&amp;publication_year=1992&amp;author=White%2CSR\" 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\">Wang, M. et al. Tensor networks meet neural networks: a survey and future perspectives. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2302.09019\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2302.09019\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2302.09019<\/a> (2023).<\/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\">Liao, H.-J., Liu, J.-G., Wang, L. &amp; Xiang, T. Differentiable programming tensor networks. Phys. Rev.X <b>9<\/b>, 031041 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 30\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Differentiable%20programming%20tensor%20networks&amp;journal=Phys.%20Rev.X&amp;volume=9&amp;publication_year=2019&amp;author=Liao%2CH-J&amp;author=Liu%2CJ-G&amp;author=Wang%2CL&amp;author=Xiang%2CT\" 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=\"31.\">\n<p class=\"c-article-references__text\" id=\"ref-CR31\">Hauru, M., Van Damme, M. &amp; Haegeman, J. Riemannian optimization of isometric tensor networks. SciPost Phys <b>10<\/b>, 040 (2021).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021ScPP...10...40H\" 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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4237716\" aria-label=\"MathSciNet reference 31\" 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 31\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Riemannian%20optimization%20of%20isometric%20tensor%20networks&amp;journal=SciPost%20Phys&amp;volume=10&amp;publication_year=2021&amp;author=Hauru%2CM&amp;author=Damme%2CM&amp;author=Haegeman%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=\"32.\">\n<p class=\"c-article-references__text\" id=\"ref-CR32\">Luchnikov, I., Ryzhov, A., Filippov, S. &amp; Ouerdane, H. QGOpt: Riemannian optimization for quantum technologies. SciPost Physics <b>10<\/b>, 079 (2021).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021ScPP...10...79L\" aria-label=\"ADS reference 32\" 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=4237773\" aria-label=\"MathSciNet reference 32\" 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 32\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=QGOpt%3A%20Riemannian%20optimization%20for%20quantum%20technologies&amp;journal=SciPost%20Physics&amp;volume=10&amp;publication_year=2021&amp;author=Luchnikov%2CI&amp;author=Ryzhov%2CA&amp;author=Filippov%2CS&amp;author=Ouerdane%2CH\" 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\">Berezutskii, A., Luchnikov, I. &amp; Fedorov, A. Simulating quantum circuits using the multi-scale entanglement renormalization ansatz. Phys. Rev. Res. <b>7<\/b>, 013063 (2025).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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=Simulating%20quantum%20circuits%20using%20the%20multi-scale%20entanglement%20renormalization%20ansatz&amp;journal=Phys.%20Rev.%20Res.&amp;volume=7&amp;publication_year=2025&amp;author=Berezutskii%2CA&amp;author=Luchnikov%2CI&amp;author=Fedorov%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=\"34.\">\n<p class=\"c-article-references__text\" id=\"ref-CR34\">Luchnikov, I., Krechetov, M. &amp; Filippov, S. Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies. New J. Phys. (2021).<\/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\">Sandvik, A. W. &amp; Vidal, G. Variational quantum Monte Carlo simulations with tensor-network states. Phys. Rev. Lett. <b>99<\/b>, 220602 (2007).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2007PhRvL..99v0602S\" aria-label=\"ADS reference 35\" 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 35\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Variational%20quantum%20Monte%20Carlo%20simulations%20with%20tensor-network%20states&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=99&amp;publication_year=2007&amp;author=Sandvik%2CAW&amp;author=Vidal%2CG\" 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\">Wang, L., Pi\u017eorn, I. &amp; Verstraete, F. Monte Carlo simulation with tensor network states. Phys. Rev. B\u2014Condensed Matter and Materials Physics <b>83<\/b>, 134421 (2011).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2011PhRvB..83m4421W\" aria-label=\"ADS reference 36\" 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 36\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Monte%20Carlo%20simulation%20with%20tensor%20network%20states&amp;journal=Phys.%20Rev.%20B%E2%80%94Condensed%20Matter%20and%20Materials%20Physics&amp;volume=83&amp;publication_year=2011&amp;author=Wang%2CL&amp;author=Pi%C5%BEorn%2CI&amp;author=Verstraete%2CF\" 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=\"37.\">\n<p class=\"c-article-references__text\" id=\"ref-CR37\">Paeckel, S. et al. Time-evolution methods for matrix-product states. Ann. Phys.\u00a0<b>411<\/b>, 167998 (2019). <b>A review of time-evolution methods on MPS.<\/b><\/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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4031374\" aria-label=\"MathSciNet reference 37\" 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 37\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Time-evolution%20methods%20for%20matrix-product%20states&amp;journal=Ann.%20Phys.&amp;volume=411&amp;publication_year=2019&amp;author=Paeckel%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=\"38.\">\n<p class=\"c-article-references__text\" id=\"ref-CR38\">Feiguin, A. E. &amp; White, S. R. Time-step targeting methods for real-time dynamics using the density matrix renormalization group. Phys. Rev. B <b>72<\/b>, 020404 (2005).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2005PhRvB..72b0404F\" aria-label=\"ADS reference 38\" 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 38\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Time-step%20targeting%20methods%20for%20real-time%20dynamics%20using%20the%20density%20matrix%20renormalization%20group&amp;journal=Phys.%20Rev.%20B&amp;volume=72&amp;publication_year=2005&amp;author=Feiguin%2CAE&amp;author=White%2CSR\" 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\">Alvarez, G., Dias da Silva, L. G., Ponce, E. &amp; Dagotto, E. Time evolution with the density-matrix renormalization-group algorithm: A generic implementation for strongly correlated electronic systems. Phys. Rev. E <b>84<\/b>, 056706 (2011).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2011PhRvE..84e6706A\" aria-label=\"ADS reference 39\" 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 39\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Time%20evolution%20with%20the%20density-matrix%20renormalization-group%20algorithm%3A%20A%20generic%20implementation%20for%20strongly%20correlated%20electronic%20systems&amp;journal=Phys.%20Rev.%20E&amp;volume=84&amp;publication_year=2011&amp;author=Alvarez%2CG&amp;author=Dias%20da%20Silva%2CLG&amp;author=Ponce%2CE&amp;author=Dagotto%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=\"40.\">\n<p class=\"c-article-references__text\" id=\"ref-CR40\">Haegeman, J. et al. Time-dependent variational principle for quantum lattices. Phys. Rev. Lett. <b>107<\/b>, 070601 (2011).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2011PhRvL.107g0601H\" 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=Time-dependent%20variational%20principle%20for%20quantum%20lattices&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=107&amp;publication_year=2011&amp;author=Haegeman%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=\"41.\">\n<p class=\"c-article-references__text\" id=\"ref-CR41\">Haegeman, J., Lubich, C., Oseledets, I., Vandereycken, B. &amp; Verstraete, F. Unifying time evolution and optimization with matrix product states. Phys. Rev. B <b>94<\/b>, 165116 (2016).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2016PhRvB..94p5116H\" aria-label=\"ADS reference 41\" 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 41\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Unifying%20time%20evolution%20and%20optimization%20with%20matrix%20product%20states&amp;journal=Phys.%20Rev.%20B&amp;volume=94&amp;publication_year=2016&amp;author=Haegeman%2CJ&amp;author=Lubich%2CC&amp;author=Oseledets%2CI&amp;author=Vandereycken%2CB&amp;author=Verstraete%2CF\" 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=\"42.\">\n<p class=\"c-article-references__text\" id=\"ref-CR42\">Ran, S.-J., Sun, Z.-Z., Fei, S.-M., Su, G. &amp; Lewenstein, M. Tensor network compressed sensing with unsupervised machine learning. Phys. Rev. Res. <b>2<\/b>, 033293 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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=Tensor%20network%20compressed%20sensing%20with%20unsupervised%20machine%20learning&amp;journal=Phys.%20Rev.%20Res.&amp;volume=2&amp;publication_year=2020&amp;author=Ran%2CS-J&amp;author=Sun%2CZ-Z&amp;author=Fei%2CS-M&amp;author=Su%2CG&amp;author=Lewenstein%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=\"43.\">\n<p class=\"c-article-references__text\" id=\"ref-CR43\">Arnborg, S., Corneil, D. G. &amp; Proskurowski, A. Complexity of finding embeddings in ak-tree. SIAM Journal on Algebraic Discrete Methods <b>8<\/b>, 277\u2013284 (1987).<\/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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=881187\" aria-label=\"MathSciNet reference 43\" 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 43\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Complexity%20of%20finding%20embeddings%20in%20ak-tree&amp;journal=SIAM%20Journal%20on%20Algebraic%20Discrete%20Methods&amp;volume=8&amp;pages=277-284&amp;publication_year=1987&amp;author=Arnborg%2CS&amp;author=Corneil%2CDG&amp;author=Proskurowski%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=\"44.\">\n<p class=\"c-article-references__text\" id=\"ref-CR44\">Pfeifer, R. N., Haegeman, J. &amp; Verstraete, F. Faster identification of optimal contraction sequences for tensor networks. Phys. Rev.E <b>90<\/b>, 033315 (2014).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2014PhRvE..90c3315P\" 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=Faster%20identification%20of%20optimal%20contraction%20sequences%20for%20tensor%20networks&amp;journal=Phys.%20Rev.E&amp;volume=90&amp;publication_year=2014&amp;author=Pfeifer%2CRN&amp;author=Haegeman%2CJ&amp;author=Verstraete%2CF\" 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\">Kourtis, S., Chamon, C., Mucciolo, E. &amp; Ruckenstein, A. Fast counting with tensor networks. SciPost Physics <b>7<\/b>, 060 (2019).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2019ScPP....7...60K\" 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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4151031\" aria-label=\"MathSciNet reference 45\" 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 45\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Fast%20counting%20with%20tensor%20networks&amp;journal=SciPost%20Physics&amp;volume=7&amp;publication_year=2019&amp;author=Kourtis%2CS&amp;author=Chamon%2CC&amp;author=Mucciolo%2CE&amp;author=Ruckenstein%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=\"46.\">\n<p class=\"c-article-references__text\" id=\"ref-CR46\">Kalachev, G., Panteleev, P., Zhou, P. &amp; Yung, M.-H. Classical sampling of random quantum circuits with bounded fidelity. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2112.15083\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2112.15083\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2112.15083<\/a> (2021).<\/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\">Morvan, A. et al. Phase transitions in random circuit sampling. Nature <b>634<\/b>, 328\u2013333 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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=Phase%20transitions%20in%20random%20circuit%20sampling&amp;journal=Nature&amp;volume=634&amp;pages=328-333&amp;publication_year=2024&amp;author=Morvan%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=\"48.\">\n<p class=\"c-article-references__text\" id=\"ref-CR48\">Meirom, E., Maron, H., Mannor, S. &amp; Chechik, G. Optimizing tensor network contraction using reinforcement learning. (2022).<\/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\">Liu, X.-Y. &amp; Zhang, Z. Classical simulation of quantum circuits using reinforcement learning: Parallel environments and benchmark (2023).<\/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\">Kalachev, G., Panteleev, P. &amp; Yung, M.-H. Multi-tensor contraction for xeb verification of quantum circuits. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2108.05665\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2108.05665\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2108.05665<\/a> (2021).<\/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\">Huang, C. et al. Classical simulation of quantum supremacy circuits. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2005.06787\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2005.06787\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2005.06787<\/a> (2020).<\/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\">Aaronson, S. &amp; Chen, L. Complexity-theoretic foundations of quantum supremacy experiments. In CCC &#8217;17: Proc. 32nd Computational Complexity Conference (CCC, 2017).<\/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\">Chen, J., Zhang, F., Huang, C., Newman, M. &amp; Shi, Y. Classical simulation of intermediate-size quantum circuits. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/1805.01450\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/1805.01450\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/1805.01450<\/a> (2018).<\/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\">Markov, I. L., Fatima, A., Isakov, S. V. &amp; Boixo, S. Quantum supremacy is both closer and farther than it appears. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/1807.10749\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/1807.10749\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/1807.10749<\/a> (2018).<\/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\">Villalonga, B. et al. A flexible high-performance simulator for verifying and benchmarking quantum circuits implemented on real hardware. npj Quantum Inf. <b>5<\/b>, 86 (2019).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2019npjQI...5...86V\" aria-label=\"ADS reference 55\" 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 55\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=A%20flexible%20high-performance%20simulator%20for%20verifying%20and%20benchmarking%20quantum%20circuits%20implemented%20on%20real%20hardware&amp;journal=npj%20Quantum%20Inf.&amp;volume=5&amp;publication_year=2019&amp;author=Villalonga%2CB\" 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\">Pednault, E. et al. Pareto-efficient quantum circuit simulation using tensor contraction deferral. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/1710.05867\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/1710.05867\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/1710.05867<\/a> (2017).<\/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\">Huang, C. et al. Efficient parallelization of tensor network contraction for simulating quantum computation. Nature Computational Science <b>1<\/b>, 578\u2013587 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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=Efficient%20parallelization%20of%20tensor%20network%20contraction%20for%20simulating%20quantum%20computation&amp;journal=Nature%20Computational%20Science&amp;volume=1&amp;pages=578-587&amp;publication_year=2021&amp;author=Huang%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=\"58.\">\n<p class=\"c-article-references__text\" id=\"ref-CR58\">Pan, F., Chen, K. &amp; Zhang, P. Solving the Sampling Problem of the Sycamore Quantum Circuits. Phys. Rev. Lett. <b>129<\/b>, 090502 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022PhRvL.129i0502P\" aria-label=\"ADS reference 58\" 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 58\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Solving%20the%20Sampling%20Problem%20of%20the%20Sycamore%20Quantum%20Circuits&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=129&amp;publication_year=2022&amp;author=Pan%2CF&amp;author=Chen%2CK&amp;author=Zhang%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=\"59.\">\n<p class=\"c-article-references__text\" id=\"ref-CR59\">Liu, Y. et al. Verifying Quantum Advantage Experiments with Multiple Amplitude Tensor Network Contraction. Phys. Rev. Lett. <b>132<\/b>, 030601 (2024).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024PhRvL.132c0601L\" aria-label=\"ADS reference 59\" 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 59\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Verifying%20Quantum%20Advantage%20Experiments%20with%20Multiple%20Amplitude%20Tensor%20Network%20Contraction&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=132&amp;publication_year=2024&amp;author=Liu%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=\"60.\">\n<p class=\"c-article-references__text\" id=\"ref-CR60\">Zhao, X.-H. et al. Leapfrogging sycamore: harnessing 1432 gpus for 7 \u00d7 faster quantum random circuit sampling. National Science Review <b>12<\/b>, nwae317 (2025).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 60\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Leapfrogging%20sycamore%3A%20harnessing%201432%20gpus%20for%207%20%C3%97%20faster%20quantum%20random%20circuit%20sampling&amp;journal=National%20Science%20Review&amp;volume=12&amp;publication_year=2025&amp;author=Zhao%2CX-H\" 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=\"61.\">\n<p class=\"c-article-references__text\" id=\"ref-CR61\">Roth, D. On the hardness of approximate reasoning. Artificial Intelligence <b>82<\/b>, 273\u2013302 (1996).<\/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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=1391064\" aria-label=\"MathSciNet reference 61\" 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 61\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=On%20the%20hardness%20of%20approximate%20reasoning&amp;journal=Artificial%20Intelligence&amp;volume=82&amp;pages=273-302&amp;publication_year=1996&amp;author=Roth%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=\"62.\">\n<p class=\"c-article-references__text\" id=\"ref-CR62\">Nishino, T. &amp; Okunishi, K. Corner transfer matrix renormalization group method. J. Phys. Soc. Japan <b>65<\/b>, 891\u2013894 (1996).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=1996JPSJ...65..891N\" aria-label=\"ADS reference 62\" 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 62\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Corner%20transfer%20matrix%20renormalization%20group%20method&amp;journal=J.%20Phys.%20Soc.%20Japan&amp;volume=65&amp;pages=891-894&amp;publication_year=1996&amp;author=Nishino%2CT&amp;author=Okunishi%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=\"63.\">\n<p class=\"c-article-references__text\" id=\"ref-CR63\">Levin, M. &amp; Nave, C. P. Tensor renormalization group approach to two-dimensional classical lattice models. Phys. Rev. Lett. <b>99<\/b>, 120601 (2007).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2007PhRvL..99l0601L\" aria-label=\"ADS reference 63\" 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 63\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tensor%20renormalization%20group%20approach%20to%20two-dimensional%20classical%20lattice%20models&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=99&amp;publication_year=2007&amp;author=Levin%2CM&amp;author=Nave%2CCP\" 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=\"64.\">\n<p class=\"c-article-references__text\" id=\"ref-CR64\">Xie, Z.-Y. et al. Coarse-graining renormalization by higher-order singular value decomposition. Phys. Rev. B\u00a0<b>86<\/b>, 045139 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2012PhRvB..86d5139X\" aria-label=\"ADS reference 64\" 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 64\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Coarse-graining%20renormalization%20by%20higher-order%20singular%20value%20decomposition&amp;journal=Phys.%20Rev.%20B&amp;volume=86&amp;publication_year=2012&amp;author=Xie%2CZ-Y\" 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=\"65.\">\n<p class=\"c-article-references__text\" id=\"ref-CR65\">Evenbly, G. &amp; Vidal, G. Tensor network renormalization. Phys. Rev. Lett. <b>115<\/b>, 180405 (2015).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2015PhRvL.115r0405E\" aria-label=\"ADS reference 65\" 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=3528219\" aria-label=\"MathSciNet reference 65\" 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 65\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tensor%20network%20renormalization&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=115&amp;publication_year=2015&amp;author=Evenbly%2CG&amp;author=Vidal%2CG\" 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=\"66.\">\n<p class=\"c-article-references__text\" id=\"ref-CR66\">Chen, J., Jiang, J., Hangleiter, D. &amp; Schuch, N. Sign problem in tensor network contraction. PRX Quantum <b>6<\/b>, 010312 (2025).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 66\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Sign%20problem%20in%20tensor%20network%20contraction&amp;journal=PRX%20Quantum&amp;volume=6&amp;publication_year=2025&amp;author=Chen%2CJ&amp;author=Jiang%2CJ&amp;author=Hangleiter%2CD&amp;author=Schuch%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=\"67.\">\n<p class=\"c-article-references__text\" id=\"ref-CR67\">Jiang, J., Chen, J., Schuch, N. &amp; Hangleiter, D. Positive bias makes tensor-network contraction tractable. In STOC &#8217;25: Proc. 57th Annual ACM Symposium on Theory of Computing (ACM, 2025).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"68.\">\n<p class=\"c-article-references__text\" id=\"ref-CR68\">Vidal, G. Class of quantum many-body states that can be efficiently simulated. Phys. Rev. Lett. <b>101<\/b>, 110501 (2008).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2008PhRvL.101k0501V\" aria-label=\"ADS reference 68\" 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 68\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Class%20of%20quantum%20many-body%20states%20that%20can%20be%20efficiently%20simulated&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=101&amp;publication_year=2008&amp;author=Vidal%2CG\" 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=\"69.\">\n<p class=\"c-article-references__text\" id=\"ref-CR69\">Verstraete, F. &amp; Cirac, J. I. Renormalization algorithms for quantum-many body systems in two and higher dimensions. arXiv preprint cond-mat\/0407066 (2004).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"70.\">\n<p class=\"c-article-references__text\" id=\"ref-CR70\">Jiang, H.-C., Weng, Z.-Y. &amp; Xiang, T. Accurate determination of tensor network state of quantum lattice models in two dimensions. Phys. Rev. Lett. <b>101<\/b>, 090603 (2008).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2008PhRvL.101i0603J\" aria-label=\"ADS reference 70\" 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 70\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Accurate%20determination%20of%20tensor%20network%20state%20of%20quantum%20lattice%20models%20in%20two%20dimensions&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=101&amp;publication_year=2008&amp;author=Jiang%2CH-C&amp;author=Weng%2CZ-Y&amp;author=Xiang%2CT\" 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=\"71.\">\n<p class=\"c-article-references__text\" id=\"ref-CR71\">Corboz, P., Jordan, J. &amp; Vidal, G. Simulation of fermionic lattice models in two dimensions with projected entangled-pair states: Next-nearest neighbor hamiltonians. Phys. Rev. B\u2014Condensed Matter and Materials Physics <b>82<\/b>, 245119 (2010).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2010PhRvB..82x5119C\" aria-label=\"ADS reference 71\" 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 71\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Simulation%20of%20fermionic%20lattice%20models%20in%20two%20dimensions%20with%20projected%20entangled-pair%20states%3A%20Next-nearest%20neighbor%20hamiltonians&amp;journal=Phys.%20Rev.%20B%E2%80%94Condensed%20Matter%20and%20Materials%20Physics&amp;volume=82&amp;publication_year=2010&amp;author=Corboz%2CP&amp;author=Jordan%2CJ&amp;author=Vidal%2CG\" 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=\"72.\">\n<p class=\"c-article-references__text\" id=\"ref-CR72\">Lubasch, M., Cirac, J. I. &amp; Banuls, M.-C. Unifying projected entangled pair state contractions. New J. Phys. <b>16<\/b>, 033014 (2014).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2014NJPh...16c3014L\" aria-label=\"ADS reference 72\" 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 72\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Unifying%20projected%20entangled%20pair%20state%20contractions&amp;journal=New%20J.%20Phys.&amp;volume=16&amp;publication_year=2014&amp;author=Lubasch%2CM&amp;author=Cirac%2CJI&amp;author=Banuls%2CM-C\" 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=\"73.\">\n<p class=\"c-article-references__text\" id=\"ref-CR73\">Pan, F., Zhou, P., Li, S. &amp; Zhang, P. Contracting arbitrary tensor networks: general approximate algorithm and applications in graphical models and quantum circuit simulations. Phys. Rev. Lett. <b>125<\/b>, 060503 (2020).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020PhRvL.125f0503P\" aria-label=\"ADS reference 73\" 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 73\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Contracting%20arbitrary%20tensor%20networks%3A%20general%20approximate%20algorithm%20and%20applications%20in%20graphical%20models%20and%20quantum%20circuit%20simulations&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=125&amp;publication_year=2020&amp;author=Pan%2CF&amp;author=Zhou%2CP&amp;author=Li%2CS&amp;author=Zhang%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=\"74.\">\n<p class=\"c-article-references__text\" id=\"ref-CR74\">Ma, L., Fishman, M., Stoudenmire, M. &amp; Solomonik, E. Approximate contraction of arbitrary tensor networks with a flexible and efficient density matrix algorithm. Quantum <b>8<\/b>, 1580 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 74\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Approximate%20contraction%20of%20arbitrary%20tensor%20networks%20with%20a%20flexible%20and%20efficient%20density%20matrix%20algorithm&amp;journal=Quantum&amp;volume=8&amp;publication_year=2024&amp;author=Ma%2CL&amp;author=Fishman%2CM&amp;author=Stoudenmire%2CM&amp;author=Solomonik%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=\"75.\">\n<p class=\"c-article-references__text\" id=\"ref-CR75\">Gray, J. &amp; Chan, G. K.-L. Hyperoptimized approximate contraction of tensor networks with arbitrary geometry. Phys. Rev.X <b>14<\/b>, 011009 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 75\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Hyperoptimized%20approximate%20contraction%20of%20tensor%20networks%20with%20arbitrary%20geometry&amp;journal=Phys.%20Rev.X&amp;volume=14&amp;publication_year=2024&amp;author=Gray%2CJ&amp;author=Chan%2CGK-L\" 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=\"76.\">\n<p class=\"c-article-references__text\" id=\"ref-CR76\">Zhou, Y., Stoudenmire, E. M. &amp; Waintal, X. What limits the simulation of quantum computers? Phys. Rev.X <b>10<\/b>, 041038 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 76\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=What%20limits%20the%20simulation%20of%20quantum%20computers%3F&amp;journal=Phys.%20Rev.X&amp;volume=10&amp;publication_year=2020&amp;author=Zhou%2CY&amp;author=Stoudenmire%2CEM&amp;author=Waintal%2CX\" 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=\"77.\">\n<p class=\"c-article-references__text\" id=\"ref-CR77\">Liu, M., Liu, J., Alexeev, Y. &amp; Jiang, L. Estimating the randomness of quantum circuit ensembles up to 50 qubits. npj Quantum Inf. <b>8<\/b>, 137 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022npjQI...8..137L\" aria-label=\"ADS reference 77\" 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 77\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Estimating%20the%20randomness%20of%20quantum%20circuit%20ensembles%20up%20to%2050%20qubits&amp;journal=npj%20Quantum%20Inf.&amp;volume=8&amp;publication_year=2022&amp;author=Liu%2CM&amp;author=Liu%2CJ&amp;author=Alexeev%2CY&amp;author=Jiang%2CL\" 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=\"78.\">\n<p class=\"c-article-references__text\" id=\"ref-CR78\">Kim, Y. et al. Evidence for the utility of quantum computing before fault tolerance. Nature <b>618<\/b>, 500\u2013505 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023Natur.618..500K\" aria-label=\"ADS reference 78\" 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 78\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Evidence%20for%20the%20utility%20of%20quantum%20computing%20before%20fault%20tolerance&amp;journal=Nature&amp;volume=618&amp;pages=500-505&amp;publication_year=2023&amp;author=Kim%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=\"79.\">\n<p class=\"c-article-references__text\" id=\"ref-CR79\">Zaletel, M. P. &amp; Pollmann, F. Isometric tensor network states in two dimensions. Phys. Rev. Lett. <b>124<\/b>, 037201 (2020).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020PhRvL.124c7201Z\" aria-label=\"ADS reference 79\" 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=4056594\" aria-label=\"MathSciNet reference 79\" 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 79\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Isometric%20tensor%20network%20states%20in%20two%20dimensions&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=124&amp;publication_year=2020&amp;author=Zaletel%2CMP&amp;author=Pollmann%2CF\" 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=\"80.\">\n<p class=\"c-article-references__text\" id=\"ref-CR80\">Tindall, J. &amp; Fishman, M. Gauging tensor networks with belief propagation. SciPost Physics <b>15<\/b>, 222 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023ScPP...15..222T\" aria-label=\"ADS reference 80\" 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=4701006\" aria-label=\"MathSciNet reference 80\" 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 80\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Gauging%20tensor%20networks%20with%20belief%20propagation&amp;journal=SciPost%20Physics&amp;volume=15&amp;publication_year=2023&amp;author=Tindall%2CJ&amp;author=Fishman%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=\"81.\">\n<p class=\"c-article-references__text\" id=\"ref-CR81\">Anand, S., Temme, K., Kandala, A. &amp; Zaletel, M. Classical benchmarking of zero noise extrapolation beyond the exactly-verifiable regime. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2306.17839\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2306.17839\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2306.17839<\/a> (2023).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"82.\">\n<p class=\"c-article-references__text\" id=\"ref-CR82\">Patra, S., Jahromi, S. S., Singh, S. &amp; Or\u00fas, R. Efficient tensor network simulation of ibm\u2019s largest quantum processors. Phys. Rev. Res. <b>6<\/b>, 013326 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 82\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Efficient%20tensor%20network%20simulation%20of%20ibm%E2%80%99s%20largest%20quantum%20processors&amp;journal=Phys.%20Rev.%20Res.&amp;volume=6&amp;publication_year=2024&amp;author=Patra%2CS&amp;author=Jahromi%2CSS&amp;author=Singh%2CS&amp;author=Or%C3%BAs%2CR\" 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=\"83.\">\n<p class=\"c-article-references__text\" id=\"ref-CR83\">Rudolph, M. S., Fontana, E., Holmes, Z. &amp; Cincio, L. Classical surrogate simulation of quantum systems with lowesa. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2308.09109\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2308.09109\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2308.09109<\/a> (2023).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"84.\">\n<p class=\"c-article-references__text\" id=\"ref-CR84\">Bouland, A., Fefferman, B., Landau, Z. &amp; Liu, Y. Noise and the frontier of quantum supremacy (2022).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"85.\">\n<p class=\"c-article-references__text\" id=\"ref-CR85\">Krovi, H. Average-case hardness of estimating probabilities of random quantum circuits with a linear scaling in the error exponent. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2206.05642\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2206.05642\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2206.05642<\/a> (2022).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"86.\">\n<p class=\"c-article-references__text\" id=\"ref-CR86\">Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature <b>574<\/b>, 505\u2013510 (2019). <b>The first experimental realization of random circuit sampling claiming quantum advantage.<\/b><\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2019Natur.574..505A\" aria-label=\"ADS reference 86\" 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 86\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20supremacy%20using%20a%20programmable%20superconducting%20processor&amp;journal=Nature&amp;volume=574&amp;pages=505-510&amp;publication_year=2019&amp;author=Arute%2CF\" 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=\"87.\">\n<p class=\"c-article-references__text\" id=\"ref-CR87\">Wu, Y. et al. Strong quantum computational advantage using a superconducting quantum processor. Phys. Rev. Lett. <b>127<\/b>, 180501 (2021).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021PhRvL.127r0501W\" aria-label=\"ADS reference 87\" 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 87\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Strong%20quantum%20computational%20advantage%20using%20a%20superconducting%20quantum%20processor&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=127&amp;publication_year=2021&amp;author=Wu%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=\"88.\">\n<p class=\"c-article-references__text\" id=\"ref-CR88\">Zhu, Q. et al. Quantum computational advantage via 60-qubit 24-cycle random circuit sampling. Science bulletin <b>67<\/b>, 240\u2013245 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022SciBu..67..240Z\" aria-label=\"ADS reference 88\" 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 88\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20computational%20advantage%20via%2060-qubit%2024-cycle%20random%20circuit%20sampling&amp;journal=Science%20bulletin&amp;volume=67&amp;pages=240-245&amp;publication_year=2022&amp;author=Zhu%2CQ\" 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=\"89.\">\n<p class=\"c-article-references__text\" id=\"ref-CR89\">Aaronson, S. &amp; Hung, S.-H. Certified randomness from quantum supremacy (2023).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"90.\">\n<p class=\"c-article-references__text\" id=\"ref-CR90\">Liu, M. et al. Certified randomness with a trapped-ion quantum processor. Nature <b>640<\/b>, 343\u2013348 (2025).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 90\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Certified%20randomness%20with%20a%20trapped-ion%20quantum%20processor&amp;journal=Nature&amp;volume=640&amp;pages=343-348&amp;publication_year=2025&amp;author=Liu%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=\"91.\">\n<p class=\"c-article-references__text\" id=\"ref-CR91\">Kaleoglu, F. et al. On the equivalence between classical position verification and certified randomness. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2410.03982\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2410.03982\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2410.03982<\/a> (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"92.\">\n<p class=\"c-article-references__text\" id=\"ref-CR92\">Amer, O. et al. Applications of certified randomness. Nat. Rev. Phys. <a href=\"https:\/\/doi.org\/10.1038\/s42254-025-00845-1\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.1038\/s42254-025-00845-1\" target=\"_blank\" rel=\"noopener\">https:\/\/doi.org\/10.1038\/s42254-025-00845-1<\/a> (2025).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"93.\">\n<p class=\"c-article-references__text\" id=\"ref-CR93\">Pednault, E., Gunnels, J., Maslov, D. &amp; Gambetta, J. On \u201cquantum supremacy\u201d. IBM Research Blog <b>21<\/b> (2019).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"94.\">\n<p class=\"c-article-references__text\" id=\"ref-CR94\">Schutski, R., Lykov, D. &amp; Oseledets, I. Adaptive algorithm for quantum circuit simulation. Phys. Rev.A <b>101<\/b>, 042335 (2020).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020PhRvA.101d2335S\" aria-label=\"ADS reference 94\" 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 94\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Adaptive%20algorithm%20for%20quantum%20circuit%20simulation&amp;journal=Phys.%20Rev.A&amp;volume=101&amp;publication_year=2020&amp;author=Schutski%2CR&amp;author=Lykov%2CD&amp;author=Oseledets%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=\"95.\">\n<p class=\"c-article-references__text\" id=\"ref-CR95\">Ayral, T. et al. Density-matrix renormalization group algorithm for simulating quantum circuits with a finite fidelity. PRX Quantum <b>4<\/b>, 020304 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023PRXQ....4b0304A\" aria-label=\"ADS reference 95\" 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 95\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Density-matrix%20renormalization%20group%20algorithm%20for%20simulating%20quantum%20circuits%20with%20a%20finite%20fidelity&amp;journal=PRX%20Quantum&amp;volume=4&amp;publication_year=2023&amp;author=Ayral%2CT\" 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=\"96.\">\n<p class=\"c-article-references__text\" id=\"ref-CR96\">Haghshenas, R. et al. Digital quantum magnetism at the frontier of classical simulations. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2503.20870\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2503.20870\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2503.20870<\/a> (2025).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"97.\">\n<p class=\"c-article-references__text\" id=\"ref-CR97\">Noh, K., Jiang, L. &amp; Fefferman, B. Efficient classical simulation of noisy random quantum circuits in one dimension. Quantum <b>4<\/b>, 318 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 97\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Efficient%20classical%20simulation%20of%20noisy%20random%20quantum%20circuits%20in%20one%20dimension&amp;journal=Quantum&amp;volume=4&amp;publication_year=2020&amp;author=Noh%2CK&amp;author=Jiang%2CL&amp;author=Fefferman%2CB\" 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=\"98.\">\n<p class=\"c-article-references__text\" id=\"ref-CR98\">Cheng, S. et al. Simulating noisy quantum circuits with matrix product density operators. Phys. Rev. Res. <b>3<\/b>, 023005 (2021).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021tpte.book.....C\" aria-label=\"ADS reference 98\" 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 98\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Simulating%20noisy%20quantum%20circuits%20with%20matrix%20product%20density%20operators&amp;journal=Phys.%20Rev.%20Res.&amp;volume=3&amp;publication_year=2021&amp;author=Cheng%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=\"99.\">\n<p class=\"c-article-references__text\" id=\"ref-CR99\">Guo, C. et al. General-purpose quantum circuit simulator with projected entangled-pair states and the quantum supremacy frontier. Phys. Rev. Lett. <b>123<\/b>, 190501 (2019).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2019PhRvL.123s0501G\" aria-label=\"ADS reference 99\" 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 99\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=General-purpose%20quantum%20circuit%20simulator%20with%20projected%20entangled-pair%20states%20and%20the%20quantum%20supremacy%20frontier&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=123&amp;publication_year=2019&amp;author=Guo%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=\"100.\">\n<p class=\"c-article-references__text\" id=\"ref-CR100\">Ellerbrock, R. &amp; Martinez, T. J. A multilayer multi-configurational approach to efficiently simulate large-scale circuit-based quantum computers on classical machines. J. Chem. Phys. <b>153<\/b>, 051101 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 100\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=A%20multilayer%20multi-configurational%20approach%20to%20efficiently%20simulate%20large-scale%20circuit-based%20quantum%20computers%20on%20classical%20machines&amp;journal=J.%20Chem.%20Phys.&amp;volume=153&amp;publication_year=2020&amp;author=Ellerbrock%2CR&amp;author=Martinez%2CTJ\" 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=\"101.\">\n<p class=\"c-article-references__text\" id=\"ref-CR101\">Dumitrescu, E. Tree tensor network approach to simulating shor\u2019s algorithm. Phys. Rev.A <b>96<\/b>, 062322 (2017).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2017PhRvA..96f2322D\" aria-label=\"ADS reference 101\" 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=3746768\" aria-label=\"MathSciNet reference 101\" 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 101\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tree%20tensor%20network%20approach%20to%20simulating%20shor%E2%80%99s%20algorithm&amp;journal=Phys.%20Rev.A&amp;volume=96&amp;publication_year=2017&amp;author=Dumitrescu%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=\"102.\">\n<p class=\"c-article-references__text\" id=\"ref-CR102\">Gao, D. et al. Establishing a new benchmark in quantum computational advantage with 105-qubit zuchongzhi 3.0 processor. Phys. Rev. Lett. <b>134<\/b>, 090601 (2025).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 102\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Establishing%20a%20new%20benchmark%20in%20quantum%20computational%20advantage%20with%20105-qubit%20zuchongzhi%203.0%20processor&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=134&amp;publication_year=2025&amp;author=Gao%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=\"103.\">\n<p class=\"c-article-references__text\" id=\"ref-CR103\">Gross, C. &amp; Bloch, I. Quantum simulations with ultracold atoms in optical lattices. Science <b>357<\/b>, 995\u20131001 (2017).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2017Sci...357..995G\" aria-label=\"ADS reference 103\" 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 103\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20simulations%20with%20ultracold%20atoms%20in%20optical%20lattices&amp;journal=Science&amp;volume=357&amp;pages=995-1001&amp;publication_year=2017&amp;author=Gross%2CC&amp;author=Bloch%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=\"104.\">\n<p class=\"c-article-references__text\" id=\"ref-CR104\">Blatt, R. &amp; Roos, C. F. Quantum simulations with trapped ions. Nature Physics <b>8<\/b>, 277\u2013284 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2012NatPh...8..277B\" aria-label=\"ADS reference 104\" 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 104\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20simulations%20with%20trapped%20ions&amp;journal=Nature%20Physics&amp;volume=8&amp;pages=277-284&amp;publication_year=2012&amp;author=Blatt%2CR&amp;author=Roos%2CCF\" 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=\"105.\">\n<p class=\"c-article-references__text\" id=\"ref-CR105\">Browaeys, A. &amp; Lahaye, T. Many-body physics with individually controlled Rydberg atoms. Nature Physics <b>16<\/b>, 132\u2013142 (2020).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020NatPh..16..132B\" aria-label=\"ADS reference 105\" 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 105\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Many-body%20physics%20with%20individually%20controlled%20Rydberg%20atoms&amp;journal=Nature%20Physics&amp;volume=16&amp;pages=132-142&amp;publication_year=2020&amp;author=Browaeys%2CA&amp;author=Lahaye%2CT\" 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=\"106.\">\n<p class=\"c-article-references__text\" id=\"ref-CR106\">Aspuru-Guzik, A. &amp; Walther, P. Photonic quantum simulators. Nature physics <b>8<\/b>, 285\u2013291 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2012NatPh...8..285A\" aria-label=\"ADS reference 106\" 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 106\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Photonic%20quantum%20simulators&amp;journal=Nature%20physics&amp;volume=8&amp;pages=285-291&amp;publication_year=2012&amp;author=Aspuru-Guzik%2CA&amp;author=Walther%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=\"107.\">\n<p class=\"c-article-references__text\" id=\"ref-CR107\">Houck, A. A., T\u00fcreci, H. E. &amp; Koch, J. On-chip quantum simulation with superconducting circuits. Nature Physics <b>8<\/b>, 292\u2013299 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2012NatPh...8..292H\" aria-label=\"ADS reference 107\" 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 107\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=On-chip%20quantum%20simulation%20with%20superconducting%20circuits&amp;journal=Nature%20Physics&amp;volume=8&amp;pages=292-299&amp;publication_year=2012&amp;author=Houck%2CAA&amp;author=T%C3%BCreci%2CHE&amp;author=Koch%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=\"108.\">\n<p class=\"c-article-references__text\" id=\"ref-CR108\">Kadowaki, T. &amp; Nishimori, H. Quantum annealing in the transverse ising model. Phys. Rev.E <b>58<\/b>, 5355 (1998).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=1998PhRvE..58.5355K\" aria-label=\"ADS reference 108\" 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 108\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20annealing%20in%20the%20transverse%20ising%20model&amp;journal=Phys.%20Rev.E&amp;volume=58&amp;publication_year=1998&amp;author=Kadowaki%2CT&amp;author=Nishimori%2CH\" 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=\"109.\">\n<p class=\"c-article-references__text\" id=\"ref-CR109\">King, A. D. et al. Beyond-classical computation in quantum simulation. Science <b>388<\/b>, 6743 (2025).<\/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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4898463\" aria-label=\"MathSciNet reference 109\" 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 109\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Beyond-classical%20computation%20in%20quantum%20simulation&amp;journal=Science&amp;volume=388&amp;publication_year=2025&amp;author=King%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=\"110.\">\n<p class=\"c-article-references__text\" id=\"ref-CR110\">Shaw, A. L. et al. Benchmarking highly entangled states on a 60-atom analogue quantum simulator. Nature <b>628<\/b>, 71\u201377 (2024).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024Natur.628...71S\" aria-label=\"ADS reference 110\" 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 110\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Benchmarking%20highly%20entangled%20states%20on%20a%2060-atom%20analogue%20quantum%20simulator&amp;journal=Nature&amp;volume=628&amp;pages=71-77&amp;publication_year=2024&amp;author=Shaw%2CAL\" 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=\"111.\">\n<p class=\"c-article-references__text\" id=\"ref-CR111\">King, A. D. et al. Comment on: \u201cDynamics of disordered quantum systems with two- and three-dimensional tensor networks\u201d. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2504.06283\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2504.06283\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2504.06283<\/a> (2025).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"112.\">\n<p class=\"c-article-references__text\" id=\"ref-CR112\">Aaronson, S. &amp; Arkhipov, A. The computational complexity of linear optics (2011). <b>Proposal of a near-term experiment for quantum advantage based on sampling, specifically boson sampling using linear optics.<\/b><\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"113.\">\n<p class=\"c-article-references__text\" id=\"ref-CR113\">Zhong, H.-S. et al. Quantum computational advantage using photons. Science <b>370<\/b>, 1460\u20131463 (2020). <b>The first experimental realization of boson sampling claiming quantum advantage.<\/b><\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020Sci...370.1460Z\" aria-label=\"ADS reference 113\" 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 113\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20computational%20advantage%20using%20photons&amp;journal=Science&amp;volume=370&amp;pages=1460-1463&amp;publication_year=2020&amp;author=Zhong%2CH-S\" 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=\"114.\">\n<p class=\"c-article-references__text\" id=\"ref-CR114\">Zhong, H.-S. et al. Phase-programmable Gaussian boson sampling using stimulated squeezed light. Phys. Rev. Lett. <b>127<\/b>, 180502 (2021).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021PhRvL.127r0502Z\" aria-label=\"ADS reference 114\" 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 114\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Phase-programmable%20Gaussian%20boson%20sampling%20using%20stimulated%20squeezed%20light&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=127&amp;publication_year=2021&amp;author=Zhong%2CH-S\" 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=\"115.\">\n<p class=\"c-article-references__text\" id=\"ref-CR115\">Madsen, L. S. et al. Quantum computational advantage with a programmable photonic processor. Nature <b>606<\/b>, 75\u201381 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022Natur.606...75M\" aria-label=\"ADS reference 115\" 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 115\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20computational%20advantage%20with%20a%20programmable%20photonic%20processor&amp;journal=Nature&amp;volume=606&amp;pages=75-81&amp;publication_year=2022&amp;author=Madsen%2CLS\" 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=\"116.\">\n<p class=\"c-article-references__text\" id=\"ref-CR116\">Deng, Y.-H. et al. Gaussian boson sampling with pseudo-photon-number-resolving detectors and quantum computational advantage. Phys. Rev. Lett. <b>131<\/b>, 150601 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023PhRvL.131o0601D\" aria-label=\"ADS reference 116\" 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 116\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Gaussian%20boson%20sampling%20with%20pseudo-photon-number-resolving%20detectors%20and%20quantum%20computational%20advantage&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=131&amp;publication_year=2023&amp;author=Deng%2CY-H\" 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=\"117.\">\n<p class=\"c-article-references__text\" id=\"ref-CR117\">Huang, H.-L., Bao, W.-S. &amp; Guo, C. Simulating the dynamics of single photons in boson sampling devices with matrix product states. Phys. Rev. A <b>100<\/b>, 032305 (2019).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2019PhRvA.100c2305H\" aria-label=\"ADS reference 117\" 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 117\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Simulating%20the%20dynamics%20of%20single%20photons%20in%20boson%20sampling%20devices%20with%20matrix%20product%20states&amp;journal=Phys.%20Rev.%20A&amp;volume=100&amp;publication_year=2019&amp;author=Huang%2CH-L&amp;author=Bao%2CW-S&amp;author=Guo%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=\"118.\">\n<p class=\"c-article-references__text\" id=\"ref-CR118\">Oh, C., Noh, K., Fefferman, B. &amp; Jiang, L. Classical simulation of lossy boson sampling using matrix product operators. Phys. Rev. A <b>104<\/b>, 022407 (2021).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021PhRvA.104b2407O\" aria-label=\"ADS reference 118\" 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=4307859\" aria-label=\"MathSciNet reference 118\" 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 118\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Classical%20simulation%20of%20lossy%20boson%20sampling%20using%20matrix%20product%20operators&amp;journal=Phys.%20Rev.%20A&amp;volume=104&amp;publication_year=2021&amp;author=Oh%2CC&amp;author=Noh%2CK&amp;author=Fefferman%2CB&amp;author=Jiang%2CL\" 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=\"119.\">\n<p class=\"c-article-references__text\" id=\"ref-CR119\">Liu, M., Oh, C., Liu, J., Jiang, L. &amp; Alexeev, Y. Simulating lossy Gaussian boson sampling with matrix-product operators. Phys. Rev. A <b>108<\/b>, 052604 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023PhRvA.108e2604L\" aria-label=\"ADS reference 119\" 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 119\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Simulating%20lossy%20Gaussian%20boson%20sampling%20with%20matrix-product%20operators&amp;journal=Phys.%20Rev.%20A&amp;volume=108&amp;publication_year=2023&amp;author=Liu%2CM&amp;author=Oh%2CC&amp;author=Liu%2CJ&amp;author=Jiang%2CL&amp;author=Alexeev%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=\"120.\">\n<p class=\"c-article-references__text\" id=\"ref-CR120\">Hamilton, C. S. et al. Gaussian boson sampling. Phys. Rev. Lett. <b>119<\/b>, 170501 (2017).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2017PhRvL.119q0501H\" aria-label=\"ADS reference 120\" 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 120\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Gaussian%20boson%20sampling&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=119&amp;publication_year=2017&amp;author=Hamilton%2CCS\" 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=\"121.\">\n<p class=\"c-article-references__text\" id=\"ref-CR121\">Quesada, N. et al. Quadratic speed-up for simulating Gaussian boson sampling. PRX Quantum <b>3<\/b>, 010306 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022PRXQ....3a0306Q\" aria-label=\"ADS reference 121\" 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 121\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quadratic%20speed-up%20for%20simulating%20Gaussian%20boson%20sampling&amp;journal=PRX%20Quantum&amp;volume=3&amp;publication_year=2022&amp;author=Quesada%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=\"122.\">\n<p class=\"c-article-references__text\" id=\"ref-CR122\">Cilluffo, D., Lorenzoni, N. &amp; Plenio, M. B. Simulating Gaussian boson sampling with tensor networks in the Heisenberg picture. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2305.11215\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2305.11215\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2305.11215<\/a> (2023).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"123.\">\n<p class=\"c-article-references__text\" id=\"ref-CR123\">Nielsen, M. A. &amp; Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2010).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"124.\">\n<p class=\"c-article-references__text\" id=\"ref-CR124\">Biamonte, J. &amp; Bergholm, V. Tensor networks in a nutshell. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/1708.00006\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/1708.00006\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/1708.00006<\/a> (2017).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"125.\">\n<p class=\"c-article-references__text\" id=\"ref-CR125\">Sch\u00f6n, C., Solano, E., Verstraete, F., Cirac, J. I. &amp; Wolf, M. M. Sequential generation of entangled multiqubit states. Phys. Rev. Lett. <b>95<\/b>, 110503 (2005). <b>This paper shows how MPS can be synthesized as quantum circuits.<\/b><\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2005PhRvL..95k0503S\" aria-label=\"ADS reference 125\" 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 125\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Sequential%20generation%20of%20entangled%20multiqubit%20states&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=95&amp;publication_year=2005&amp;author=Sch%C3%B6n%2CC&amp;author=Solano%2CE&amp;author=Verstraete%2CF&amp;author=Cirac%2CJI&amp;author=Wolf%2CMM\" 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=\"126.\">\n<p class=\"c-article-references__text\" id=\"ref-CR126\">Perez-Garcia, D., Verstraete, F., Wolf, M. M. &amp; Cirac, J. I. Matrix product state representations. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/quant-ph\/0608197\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/quant-ph\/0608197\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/quant-ph\/0608197<\/a> (2006).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"127.\">\n<p class=\"c-article-references__text\" id=\"ref-CR127\">Ran, S.-J. Encoding of matrix product states into quantum circuits of one- and two-qubit gates. Phys. Rev. A <b>101<\/b>, 032310 (2020).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020PhRvA.101c2310R\" aria-label=\"ADS reference 127\" 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=4086397\" aria-label=\"MathSciNet reference 127\" 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 127\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Encoding%20of%20matrix%20product%20states%20into%20quantum%20circuits%20of%20one-%20and%20two-qubit%20gates&amp;journal=Phys.%20Rev.%20A&amp;volume=101&amp;publication_year=2020&amp;author=Ran%2CS-J\" 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=\"128.\">\n<p class=\"c-article-references__text\" id=\"ref-CR128\">Wei, Z.-Y., Malz, D. &amp; Cirac, J. I. Efficient adiabatic preparation of tensor network states. Phys. Rev. Res. <b>5<\/b>, L022037 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 128\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Efficient%20adiabatic%20preparation%20of%20tensor%20network%20states&amp;journal=Phys.%20Rev.%20Res.&amp;volume=5&amp;publication_year=2023&amp;author=Wei%2CZ-Y&amp;author=Malz%2CD&amp;author=Cirac%2CJI\" 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=\"129.\">\n<p class=\"c-article-references__text\" id=\"ref-CR129\">Malz, D., Styliaris, G., Wei, Z.-Y. &amp; Cirac, J. I. Preparation of matrix product states with log-depth quantum circuits. Phys. Rev. Lett. <b>132<\/b>, 040404 (2024).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024PhRvL.132d0404M\" aria-label=\"ADS reference 129\" 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=4709261\" aria-label=\"MathSciNet reference 129\" 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 129\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Preparation%20of%20matrix%20product%20states%20with%20log-depth%20quantum%20circuits&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=132&amp;publication_year=2024&amp;author=Malz%2CD&amp;author=Styliaris%2CG&amp;author=Wei%2CZ-Y&amp;author=Cirac%2CJI\" 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=\"130.\">\n<p class=\"c-article-references__text\" id=\"ref-CR130\">Haghshenas, R., Gray, J., Potter, A. C. &amp; Chan, G. K.-L. Variational power of quantum circuit tensor networks. Phys. Rev. X <b>12<\/b>, 011047 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 130\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Variational%20power%20of%20quantum%20circuit%20tensor%20networks&amp;journal=Phys.%20Rev.%20X&amp;volume=12&amp;publication_year=2022&amp;author=Haghshenas%2CR&amp;author=Gray%2CJ&amp;author=Potter%2CAC&amp;author=Chan%2CGK-L\" 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=\"131.\">\n<p class=\"c-article-references__text\" id=\"ref-CR131\">Foss-Feig, M. et al. Holographic quantum algorithms for simulating correlated spin systems. Phys. Rev. Res. <b>3<\/b>, 033002 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 131\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Holographic%20quantum%20algorithms%20for%20simulating%20correlated%20spin%20systems&amp;journal=Phys.%20Rev.%20Res.&amp;volume=3&amp;publication_year=2021&amp;author=Foss-Feig%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=\"132.\">\n<p class=\"c-article-references__text\" id=\"ref-CR132\">Barratt, F. et al. Parallel quantum simulation of large systems on small NISQ computers. npj Quantum Inf. <b>7<\/b>, 79 (2021).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021npjQI...7...79B\" aria-label=\"ADS reference 132\" 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 132\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Parallel%20quantum%20simulation%20of%20large%20systems%20on%20small%20NISQ%20computers&amp;journal=npj%20Quantum%20Inf.&amp;volume=7&amp;publication_year=2021&amp;author=Barratt%2CF\" 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=\"133.\">\n<p class=\"c-article-references__text\" id=\"ref-CR133\">Smith, A., Jobst, B., Green, A. G. &amp; Pollmann, F. Crossing a topological phase transition with a quantum computer. Phys. Rev. Res. <b>4<\/b>, L022020 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 133\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Crossing%20a%20topological%20phase%20transition%20with%20a%20quantum%20computer&amp;journal=Phys.%20Rev.%20Res.&amp;volume=4&amp;publication_year=2022&amp;author=Smith%2CA&amp;author=Jobst%2CB&amp;author=Green%2CAG&amp;author=Pollmann%2CF\" 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=\"134.\">\n<p class=\"c-article-references__text\" id=\"ref-CR134\">Meth, M. et al. Probing phases of quantum matter with an ion-trap tensor-network quantum eigensolver. Phys. Rev. X <b>12<\/b>, 041035 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 134\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Probing%20phases%20of%20quantum%20matter%20with%20an%20ion-trap%20tensor-network%20quantum%20eigensolver&amp;journal=Phys.%20Rev.%20X&amp;volume=12&amp;publication_year=2022&amp;author=Meth%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=\"135.\">\n<p class=\"c-article-references__text\" id=\"ref-CR135\">Lin, S.-H., Dilip, R., Green, A. G., Smith, A. &amp; Pollmann, F. Real-and imaginary-time evolution with compressed quantum circuits. PRX Quantum <b>2<\/b>, 010342 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 135\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Real-and%20imaginary-time%20evolution%20with%20compressed%20quantum%20circuits&amp;journal=PRX%20Quantum&amp;volume=2&amp;publication_year=2021&amp;author=Lin%2CS-H&amp;author=Dilip%2CR&amp;author=Green%2CAG&amp;author=Smith%2CA&amp;author=Pollmann%2CF\" 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=\"136.\">\n<p class=\"c-article-references__text\" id=\"ref-CR136\">Haghshenas, R., O\u2019Rourke, M. J. &amp; Chan, G. K.-L. Conversion of projected entangled pair states into a canonical form. Phys. Rev. B <b>100<\/b>, 054404 (2019).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2019PhRvB.100e4404H\" aria-label=\"ADS reference 136\" 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 136\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Conversion%20of%20projected%20entangled%20pair%20states%20into%20a%20canonical%20form&amp;journal=Phys.%20Rev.%20B&amp;volume=100&amp;publication_year=2019&amp;author=Haghshenas%2CR&amp;author=O%E2%80%99Rourke%2CMJ&amp;author=Chan%2CGK-L\" 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=\"137.\">\n<p class=\"c-article-references__text\" id=\"ref-CR137\">Wei, Z.-Y., Malz, D. &amp; Cirac, J. I. Sequential generation of projected entangled-pair states. Phys. Rev. Lett. <b>128<\/b>, 010607 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022PhRvL.128a0607W\" aria-label=\"ADS reference 137\" 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=4387798\" aria-label=\"MathSciNet reference 137\" 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 137\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Sequential%20generation%20of%20projected%20entangled-pair%20states&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=128&amp;publication_year=2022&amp;author=Wei%2CZ-Y&amp;author=Malz%2CD&amp;author=Cirac%2CJI\" 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=\"138.\">\n<p class=\"c-article-references__text\" id=\"ref-CR138\">Kim, I. H. &amp; Swingle, B. Robust entanglement renormalization on a noisy quantum computer. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/1711.07500\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/1711.07500\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/1711.07500<\/a> (2017).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"139.\">\n<p class=\"c-article-references__text\" id=\"ref-CR139\">Mart\u00edn, E. C., Plekhanov, K. &amp; Lubasch, M. Barren plateaus in quantum tensor network optimization. Quantum <b>7<\/b>, 974 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 139\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Barren%20plateaus%20in%20quantum%20tensor%20network%20optimization&amp;journal=Quantum&amp;volume=7&amp;publication_year=2023&amp;author=Mart%C3%ADn%2CEC&amp;author=Plekhanov%2CK&amp;author=Lubasch%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=\"140.\">\n<p class=\"c-article-references__text\" id=\"ref-CR140\">Barthel, T. &amp; Miao, Q. Absence of barren plateaus and scaling of gradients in the energy optimization of isometric tensor network states. Commun. Math. Phys. <b>406<\/b>, 86 (2025).<\/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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4881034\" aria-label=\"MathSciNet reference 140\" 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 140\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Absence%20of%20barren%20plateaus%20and%20scaling%20of%20gradients%20in%20the%20energy%20optimization%20of%20isometric%20tensor%20network%20states&amp;journal=Commun.%20Math.%20Phys.&amp;volume=406&amp;publication_year=2025&amp;author=Barthel%2CT&amp;author=Miao%2CQ\" 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=\"141.\">\n<p class=\"c-article-references__text\" id=\"ref-CR141\">Sewell, T. J. &amp; Jordan, S. P. Preparing renormalization group fixed points on NISQ hardware. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2109.09787\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2109.09787\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2109.09787<\/a> (2021).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"142.\">\n<p class=\"c-article-references__text\" id=\"ref-CR142\">Anand, S., Hauschild, J., Zhang, Y., Potter, A. C. &amp; Zaletel, M. P. Holographic quantum simulation of entanglement renormalization circuits. PRX Quantum <b>4<\/b>, 030334 (2023). <b>This paper uses quantum circuits derived from MERA and uses a holographic encoding to simulate quantum systems larger than the quantum device.<\/b><\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023PRXQ....4c0334A\" aria-label=\"ADS reference 142\" 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 142\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Holographic%20quantum%20simulation%20of%20entanglement%20renormalization%20circuits&amp;journal=PRX%20Quantum&amp;volume=4&amp;publication_year=2023&amp;author=Anand%2CS&amp;author=Hauschild%2CJ&amp;author=Zhang%2CY&amp;author=Potter%2CAC&amp;author=Zaletel%2CMP\" 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=\"143.\">\n<p class=\"c-article-references__text\" id=\"ref-CR143\">Haghshenas, R. et al. Probing critical states of matter on a digital quantum computer. Phys. Rev. Lett. <b>133<\/b>, 266502 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 143\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Probing%20critical%20states%20of%20matter%20on%20a%20digital%20quantum%20computer&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=133&amp;publication_year=2024&amp;author=Haghshenas%2CR\" 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=\"144.\">\n<p class=\"c-article-references__text\" id=\"ref-CR144\">Miao, Q., Wang, T., Brown, K. R., Barthel, T. &amp; Cetina, M. Probing entanglement scaling across a quantum phase transition on a quantum computer. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2412.18602\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2412.18602\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2412.18602<\/a> (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"145.\">\n<p class=\"c-article-references__text\" id=\"ref-CR145\">Zhang, Y., Gopalakrishnan, S. &amp; Styliaris, G. Characterizing matrix-product states and projected entangled-pair states preparable via measurement and feedback. PRX Quantum <b>5<\/b>, 040304 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 145\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Characterizing%20matrix-product%20states%20and%20projected%20entangled-pair%20states%20preparable%20via%20measurement%20and%20feedback&amp;journal=PRX%20Quantum&amp;volume=5&amp;publication_year=2024&amp;author=Zhang%2CY&amp;author=Gopalakrishnan%2CS&amp;author=Styliaris%2CG\" 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=\"146.\">\n<p class=\"c-article-references__text\" id=\"ref-CR146\">Larsen, J. B., Grace, M. D., Baczewski, A. D. &amp; Magann, A. B. Feedback-based quantum algorithms for ground state preparation. Phys. Rev. Res. <b>6<\/b>, 033336 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 146\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Feedback-based%20quantum%20algorithms%20for%20ground%20state%20preparation&amp;journal=Phys.%20Rev.%20Res.&amp;volume=6&amp;publication_year=2024&amp;author=Larsen%2CJB&amp;author=Grace%2CMD&amp;author=Baczewski%2CAD&amp;author=Magann%2CAB\" 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=\"147.\">\n<p class=\"c-article-references__text\" id=\"ref-CR147\">Stephen, D. T. &amp; Hart, O. Preparing matrix product states via fusion: constraints and extensions. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2404.16360\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2404.16360\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2404.16360<\/a> (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"148.\">\n<p class=\"c-article-references__text\" id=\"ref-CR148\">Smith, K. C., Crane, E., Wiebe, N. &amp; Girvin, S. Deterministic constant-depth preparation of the AKLT state on a quantum processor using fusion measurements. PRX Quantum <b>4<\/b>, 020315 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023PRXQ....4b0315S\" aria-label=\"ADS reference 148\" 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 148\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Deterministic%20constant-depth%20preparation%20of%20the%20AKLT%20state%20on%20a%20quantum%20processor%20using%20fusion%20measurements&amp;journal=PRX%20Quantum&amp;volume=4&amp;publication_year=2023&amp;author=Smith%2CKC&amp;author=Crane%2CE&amp;author=Wiebe%2CN&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=\"149.\">\n<p class=\"c-article-references__text\" id=\"ref-CR149\">Sahay, R. &amp; Verresen, R. Classifying one-dimensional quantum states prepared by a single round of measurements. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2404.16753\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2404.16753\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2404.16753<\/a> (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"150.\">\n<p class=\"c-article-references__text\" id=\"ref-CR150\">Ben-Dov, M., Shnaiderov, D., Makmal, A. &amp; Dalla Torre, E. G. Approximate encoding of quantum states using shallow circuits. npj Quantum Inf. <b>10<\/b>, 65 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 150\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Approximate%20encoding%20of%20quantum%20states%20using%20shallow%20circuits&amp;journal=npj%20Quantum%20Inf.&amp;volume=10&amp;publication_year=2024&amp;author=Ben-Dov%2CM&amp;author=Shnaiderov%2CD&amp;author=Makmal%2CA&amp;author=Dalla%20Torre%2CEG\" 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=\"151.\">\n<p class=\"c-article-references__text\" id=\"ref-CR151\">Rudolph, M. S., Chen, J., Miller, J., Acharya, A. &amp; Perdomo-Ortiz, A. Decomposition of matrix product states into shallow quantum circuits. Quantum Sci. Technol. <b>9<\/b>, 015012 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024QS%26T....9a5012R\" aria-label=\"ADS reference 151\" 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 151\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Decomposition%20of%20matrix%20product%20states%20into%20shallow%20quantum%20circuits&amp;journal=Quantum%20Sci.%20Technol.&amp;volume=9&amp;publication_year=2023&amp;author=Rudolph%2CMS&amp;author=Chen%2CJ&amp;author=Miller%2CJ&amp;author=Acharya%2CA&amp;author=Perdomo-Ortiz%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=\"152.\">\n<p class=\"c-article-references__text\" id=\"ref-CR152\">Melnikov, A. A., Termanova, A. A., Dolgov, S. V., Neukart, F. &amp; Perelshtein, M. Quantum state preparation using tensor networks. Quantum Sci. Technol. <b>8<\/b>, 035027 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023QS%26T....8c5027M\" aria-label=\"ADS reference 152\" 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 152\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20state%20preparation%20using%20tensor%20networks&amp;journal=Quantum%20Sci.%20Technol.&amp;volume=8&amp;publication_year=2023&amp;author=Melnikov%2CAA&amp;author=Termanova%2CAA&amp;author=Dolgov%2CSV&amp;author=Neukart%2CF&amp;author=Perelshtein%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=\"153.\">\n<p class=\"c-article-references__text\" id=\"ref-CR153\">Termanova, A. et al. Tensor quantum programming. New J. Phys. <b>26<\/b>, 123019 (2024).<\/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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4853712\" aria-label=\"MathSciNet reference 153\" 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 153\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tensor%20quantum%20programming&amp;journal=New%20J.%20Phys.&amp;volume=26&amp;publication_year=2024&amp;author=Termanova%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=\"154.\">\n<p class=\"c-article-references__text\" id=\"ref-CR154\">Jaderberg, B. et al. Variational preparation of normal matrix product states on quantum computers. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2503.09683\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2503.09683\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2503.09683<\/a> (2025).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"155.\">\n<p class=\"c-article-references__text\" id=\"ref-CR155\">Kukliansky, A., Younis, E., Cincio, L. &amp; Iancu, C. QFactor: a domain-specific optimizer for quantum circuit instantiation. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2306.08152\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2306.08152\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2306.08152<\/a> (2023).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"156.\">\n<p class=\"c-article-references__text\" id=\"ref-CR156\">Lidar, D. A. &amp; Brun, T. A. Quantum Error Correction (Cambridge Univ. Press, 2013).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"157.\">\n<p class=\"c-article-references__text\" id=\"ref-CR157\">Ferris, A. J. &amp; Poulin, D. Tensor networks and quantum error correction. Phys. Rev. Lett. <b>113<\/b>, 030501 (2014).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2014PhRvL.113c0501F\" aria-label=\"ADS reference 157\" 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 157\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tensor%20networks%20and%20quantum%20error%20correction&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=113&amp;publication_year=2014&amp;author=Ferris%2CAJ&amp;author=Poulin%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=\"158.\">\n<p class=\"c-article-references__text\" id=\"ref-CR158\">Farrelly, T., Harris, R. J., McMahon, N. A. &amp; Stace, T. M. Tensor-network codes. Phys. Rev. Lett. <b>127<\/b>, 040507 (2021). <b>Introduces TN codes with a natural TN decoder.<\/b><\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021PhRvL.127d0507F\" aria-label=\"ADS reference 158\" 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=4304702\" aria-label=\"MathSciNet reference 158\" 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 158\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tensor-network%20codes&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=127&amp;publication_year=2021&amp;author=Farrelly%2CT&amp;author=Harris%2CRJ&amp;author=McMahon%2CNA&amp;author=Stace%2CTM\" 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=\"159.\">\n<p class=\"c-article-references__text\" id=\"ref-CR159\">Farrelly, T., Tuckett, D. K. &amp; Stace, T. M. Local tensor-network codes. New J. Phys. <b>24<\/b>, 043015 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022NJPh...24d3015F\" aria-label=\"ADS reference 159\" 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=4430786\" aria-label=\"MathSciNet reference 159\" 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 159\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Local%20tensor-network%20codes&amp;journal=New%20J.%20Phys.&amp;volume=24&amp;publication_year=2022&amp;author=Farrelly%2CT&amp;author=Tuckett%2CDK&amp;author=Stace%2CTM\" 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=\"160.\">\n<p class=\"c-article-references__text\" id=\"ref-CR160\">Cao, C. &amp; Lackey, B. Quantum lego: Building quantum error correction codes from tensor networks. PRX Quantum <b>3<\/b>, 020332 (2022). <b>Generalizes code concatenation with TNs, dubbed \u2018quantum Lego\u2019.<\/b><\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022PRXQ....3b0332C\" aria-label=\"ADS reference 160\" 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 160\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20lego%3A%20Building%20quantum%20error%20correction%20codes%20from%20tensor%20networks&amp;journal=PRX%20Quantum&amp;volume=3&amp;publication_year=2022&amp;author=Cao%2CC&amp;author=Lackey%2CB\" 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=\"161.\">\n<p class=\"c-article-references__text\" id=\"ref-CR161\">Cao, C., Gullans, M. J., Lackey, B. &amp; Wang, Z. Quantum Lego expansion pack: enumerators from tensor networks. PRX Quantum <b>5<\/b>, 030313 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 161\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20Lego%20expansion%20pack%3A%20enumerators%20from%20tensor%20networks&amp;journal=PRX%20Quantum&amp;volume=5&amp;publication_year=2024&amp;author=Cao%2CC&amp;author=Gullans%2CMJ&amp;author=Lackey%2CB&amp;author=Wang%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=\"162.\">\n<p class=\"c-article-references__text\" id=\"ref-CR162\">Fan, J. et al. LEGO_HQEC: a software tool for analyzing holographic quantum codes. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2410.22861\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2410.22861\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2410.22861<\/a> (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"163.\">\n<p class=\"c-article-references__text\" id=\"ref-CR163\">Pastawski, F., Yoshida, B., Harlow, D. &amp; Preskill, J. Holographic quantum error-correcting codes: toy models for the bulk\/boundary correspondence. J. High Energy Phys. <b>2015<\/b>, 149 (2015).<\/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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=3370186\" aria-label=\"MathSciNet reference 163\" 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 163\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Holographic%20quantum%20error-correcting%20codes%3A%20toy%20models%20for%20the%20bulk%2Fboundary%20correspondence&amp;journal=J.%20High%20Energy%20Phys.&amp;volume=2015&amp;publication_year=2015&amp;author=Pastawski%2CF&amp;author=Yoshida%2CB&amp;author=Harlow%2CD&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=\"164.\">\n<p class=\"c-article-references__text\" id=\"ref-CR164\">Jahn, A. &amp; Eisert, J. Holographic tensor network models and quantum error correction: a topical review. Quantum Sci. Technol. <b>6<\/b>, 033002 (2021).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021QS%26T....6c3002J\" aria-label=\"ADS reference 164\" 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 164\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Holographic%20tensor%20network%20models%20and%20quantum%20error%20correction%3A%20a%20topical%20review&amp;journal=Quantum%20Sci.%20Technol.&amp;volume=6&amp;publication_year=2021&amp;author=Jahn%2CA&amp;author=Eisert%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=\"165.\">\n<p class=\"c-article-references__text\" id=\"ref-CR165\">Steinberg, M., Feld, S. &amp; Jahn, A. Holographic codes from hyperinvariant tensor networks. Nat. Commun. <b>14<\/b>, 7314 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023NatCo..14.7314S\" aria-label=\"ADS reference 165\" 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 165\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Holographic%20codes%20from%20hyperinvariant%20tensor%20networks&amp;journal=Nat.%20Commun.&amp;volume=14&amp;publication_year=2023&amp;author=Steinberg%2CM&amp;author=Feld%2CS&amp;author=Jahn%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=\"166.\">\n<p class=\"c-article-references__text\" id=\"ref-CR166\">Shen, R., Wang, Y. &amp; Cao, C. Quantum Lego and XP stabilizer codes. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2310.19538\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2310.19538\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2310.19538<\/a> (2023).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"167.\">\n<p class=\"c-article-references__text\" id=\"ref-CR167\">Schotte, A., Zhu, G., Burgelman, L. &amp; Verstraete, F. Quantum error correction thresholds for the universal Fibonacci Turaev-Viro code. Phys. Rev. X <b>12<\/b>, 021012 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 167\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20error%20correction%20thresholds%20for%20the%20universal%20Fibonacci%20Turaev-Viro%20code&amp;journal=Phys.%20Rev.%20X&amp;volume=12&amp;publication_year=2022&amp;author=Schotte%2CA&amp;author=Zhu%2CG&amp;author=Burgelman%2CL&amp;author=Verstraete%2CF\" 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=\"168.\">\n<p class=\"c-article-references__text\" id=\"ref-CR168\">Bettaque, V. &amp; Swingle, B. NoRA: a tensor network ansatz for volume-law entangled equilibrium states of highly connected Hamiltonians. Quantum <b>8<\/b>, 1362 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 168\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=NoRA%3A%20a%20tensor%20network%20ansatz%20for%20volume-law%20entangled%20equilibrium%20states%20of%20highly%20connected%20Hamiltonians&amp;journal=Quantum&amp;volume=8&amp;publication_year=2024&amp;author=Bettaque%2CV&amp;author=Swingle%2CB\" 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=\"169.\">\n<p class=\"c-article-references__text\" id=\"ref-CR169\">Cao, C. &amp; Lackey, B. Approximate Bacon\u2013Shor code and holography. J. High Energy Phys. <b>2021<\/b>, 127 (2021).<\/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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4301677\" aria-label=\"MathSciNet reference 169\" 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 169\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Approximate%20Bacon%E2%80%93Shor%20code%20and%20holography&amp;journal=J.%20High%20Energy%20Phys.&amp;volume=2021&amp;publication_year=2021&amp;author=Cao%2CC&amp;author=Lackey%2CB\" 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=\"170.\">\n<p class=\"c-article-references__text\" id=\"ref-CR170\">Steinberg, M. et al. Far from perfect: quantum error correction with (hyperinvariant) Evenbly codes. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2407.11926\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2407.11926\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2407.11926<\/a> (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"171.\">\n<p class=\"c-article-references__text\" id=\"ref-CR171\">Pastawski, F. &amp; Preskill, J. Code properties from holographic geometries. Phys. Rev. X <b>7<\/b>, 021022 (2017).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 171\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Code%20properties%20from%20holographic%20geometries&amp;journal=Phys.%20Rev.%20X&amp;volume=7&amp;publication_year=2017&amp;author=Pastawski%2CF&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=\"172.\">\n<p class=\"c-article-references__text\" id=\"ref-CR172\">Su, V. P. et al. Discovery of optimal quantum error correcting codes via reinforcement learning. Phys. Rev. Appl. <b>23<\/b>, 034048 (2025).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 172\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Discovery%20of%20optimal%20quantum%20error%20correcting%20codes%20via%20reinforcement%20learning&amp;journal=Phys.%20Rev.%20Appl.&amp;volume=23&amp;publication_year=2025&amp;author=Su%2CVP\" 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=\"173.\">\n<p class=\"c-article-references__text\" id=\"ref-CR173\">Mauron, C., Farrelly, T. &amp; Stace, T. M. Optimization of tensor network codes with reinforcement learning. New J. Phys. <b>26<\/b>, 023024 (2024).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024NJPh...26b3024M\" aria-label=\"ADS reference 173\" 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=4714976\" aria-label=\"MathSciNet reference 173\" 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 173\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Optimization%20of%20tensor%20network%20codes%20with%20reinforcement%20learning&amp;journal=New%20J.%20Phys.&amp;volume=26&amp;publication_year=2024&amp;author=Mauron%2CC&amp;author=Farrelly%2CT&amp;author=Stace%2CTM\" 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=\"174.\">\n<p class=\"c-article-references__text\" id=\"ref-CR174\">Bravyi, S., Suchara, M. &amp; Vargo, A. Efficient algorithms for maximum likelihood decoding in the surface code. Phys. Rev. A <b>90<\/b>, 032326 (2014).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2014PhRvA..90c2326B\" aria-label=\"ADS reference 174\" 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 174\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Efficient%20algorithms%20for%20maximum%20likelihood%20decoding%20in%20the%20surface%20code&amp;journal=Phys.%20Rev.%20A&amp;volume=90&amp;publication_year=2014&amp;author=Bravyi%2CS&amp;author=Suchara%2CM&amp;author=Vargo%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=\"175.\">\n<p class=\"c-article-references__text\" id=\"ref-CR175\">Chubb, C. T. &amp; Flammia, S. T. Statistical mechanical models for quantum codes with correlated noise. Ann. Inst. Henri Poincar\u00e9 D <b>8<\/b>, 269\u2013321 (2021). <b>Introduces a mapping of any stabilizer code to a statistical-mechanical model, connecting the optimal threshold of the QEC code to a phase transition.<\/b><\/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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4261673\" aria-label=\"MathSciNet reference 175\" 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 175\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Statistical%20mechanical%20models%20for%20quantum%20codes%20with%20correlated%20noise&amp;journal=Ann.%20Inst.%20Henri%20Poincar%C3%A9%20D&amp;volume=8&amp;pages=269-321&amp;publication_year=2021&amp;author=Chubb%2CCT&amp;author=Flammia%2CST\" 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=\"176.\">\n<p class=\"c-article-references__text\" id=\"ref-CR176\">Darmawan, A. S., Nakata, Y., Tamiya, S. &amp; Yamasaki, H. Low-depth random Clifford circuits for quantum coding against Pauli noise using a tensor-network decoder. Phys. Rev. Res. <b>6<\/b>, 023055 (2025).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 176\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Low-depth%20random%20Clifford%20circuits%20for%20quantum%20coding%20against%20Pauli%20noise%20using%20a%20tensor-network%20decoder&amp;journal=Phys.%20Rev.%20Res.&amp;volume=6&amp;publication_year=2025&amp;author=Darmawan%2CAS&amp;author=Nakata%2CY&amp;author=Tamiya%2CS&amp;author=Yamasaki%2CH\" 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=\"177.\">\n<p class=\"c-article-references__text\" id=\"ref-CR177\">Farrelly, T., Milicevic, N., Harris, R. J., McMahon, N. A. &amp; Stace, T. M. Parallel decoding of multiple logical qubits in tensor-network codes. Phys. Rev. A <b>105<\/b>, 052446 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022PhRvA.105e2446F\" aria-label=\"ADS reference 177\" 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=4442532\" aria-label=\"MathSciNet reference 177\" 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 177\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Parallel%20decoding%20of%20multiple%20logical%20qubits%20in%20tensor-network%20codes&amp;journal=Phys.%20Rev.%20A&amp;volume=105&amp;publication_year=2022&amp;author=Farrelly%2CT&amp;author=Milicevic%2CN&amp;author=Harris%2CRJ&amp;author=McMahon%2CNA&amp;author=Stace%2CTM\" 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=\"178.\">\n<p class=\"c-article-references__text\" id=\"ref-CR178\">Google Quantum AI. Suppressing quantum errors by scaling a surface code logical qubit. Nature <b>614<\/b>, 676\u2013681 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023Natur.614..676G\" aria-label=\"ADS reference 178\" 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 178\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Suppressing%20quantum%20errors%20by%20scaling%20a%20surface%20code%20logical%20qubit&amp;journal=Nature&amp;volume=614&amp;pages=676-681&amp;publication_year=2023\" 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=\"179.\">\n<p class=\"c-article-references__text\" id=\"ref-CR179\">Piveteau, C., Chubb, C. T. &amp; Renes, J. M. Tensor-network decoding beyond 2D. PRX Quantum <b>5<\/b>, 040303 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 179\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tensor-network%20decoding%20beyond%202D&amp;journal=PRX%20Quantum&amp;volume=5&amp;publication_year=2024&amp;author=Piveteau%2CC&amp;author=Chubb%2CCT&amp;author=Renes%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=\"180.\">\n<p class=\"c-article-references__text\" id=\"ref-CR180\">Shutty, N., Newman, M. &amp; Villalonga, B. Efficient near-optimal decoding of the surface code through ensembling. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2401.12434\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2401.12434\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2401.12434<\/a> (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"181.\">\n<p class=\"c-article-references__text\" id=\"ref-CR181\">Kukliansky, A. &amp; Lackey, B. Quantum circuit tensors and enumerators with applications to quantum fault tolerance. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2405.19643\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2405.19643\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2405.19643<\/a> (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"182.\">\n<p class=\"c-article-references__text\" id=\"ref-CR182\">Cao, C. &amp; Lackey, B. Quantum weight enumerators and tensor networks. IEEE Trans. Inf. Theory <b>70<\/b>, 3512\u20133528 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023ITED...70.3504C\" aria-label=\"ADS reference 182\" 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=4740794\" aria-label=\"MathSciNet reference 182\" 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 182\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20weight%20enumerators%20and%20tensor%20networks&amp;journal=IEEE%20Trans.%20Inf.%20Theory&amp;volume=70&amp;pages=3512-3528&amp;publication_year=2023&amp;author=Cao%2CC&amp;author=Lackey%2CB\" 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=\"183.\">\n<p class=\"c-article-references__text\" id=\"ref-CR183\">Harris, R. J., McMahon, N. A., Brennen, G. K. &amp; Stace, T. M. Calderbank-shor-steane holographic quantum error-correcting codes. Phys. Rev. A <b>98<\/b>, 052301 (2018).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2018PhRvA..98e2301H\" aria-label=\"ADS reference 183\" 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 183\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Calderbank-shor-steane%20holographic%20quantum%20error-correcting%20codes&amp;journal=Phys.%20Rev.%20A&amp;volume=98&amp;publication_year=2018&amp;author=Harris%2CRJ&amp;author=McMahon%2CNA&amp;author=Brennen%2CGK&amp;author=Stace%2CTM\" 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=\"184.\">\n<p class=\"c-article-references__text\" id=\"ref-CR184\">Harris, R. J., Coupe, E., McMahon, N. A., Brennen, G. K. &amp; Stace, T. M. Decoding holographic codes with an integer optimization decoder. Phys. Rev. A <b>102<\/b>, 062417 (2020).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020PhRvA.102f2417H\" aria-label=\"ADS reference 184\" 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=4199974\" aria-label=\"MathSciNet reference 184\" 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 184\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Decoding%20holographic%20codes%20with%20an%20integer%20optimization%20decoder&amp;journal=Phys.%20Rev.%20A&amp;volume=102&amp;publication_year=2020&amp;author=Harris%2CRJ&amp;author=Coupe%2CE&amp;author=McMahon%2CNA&amp;author=Brennen%2CGK&amp;author=Stace%2CTM\" 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=\"185.\">\n<p class=\"c-article-references__text\" id=\"ref-CR185\">Bao, N. &amp; Naskar, J. Code properties of the holographic Sierpinski triangle. Phys. Rev. D <b>106<\/b>, 126006 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022PhRvD.106l6006B\" aria-label=\"ADS reference 185\" 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=4535370\" aria-label=\"MathSciNet reference 185\" 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 185\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Code%20properties%20of%20the%20holographic%20Sierpinski%20triangle&amp;journal=Phys.%20Rev.%20D&amp;volume=106&amp;publication_year=2022&amp;author=Bao%2CN&amp;author=Naskar%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=\"186.\">\n<p class=\"c-article-references__text\" id=\"ref-CR186\">Fan, J., Steinberg, M., Jahn, A., Cao, C. &amp; Feld, S. Overcoming the zero-rate hashing bound with holographic quantum error correction. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2408.06232\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2408.06232\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2408.06232<\/a> (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"187.\">\n<p class=\"c-article-references__text\" id=\"ref-CR187\">Tuckett, D. K. et al. Tailoring surface codes for highly biased noise. Phys. Rev. X <b>9<\/b>, 041031 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 187\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tailoring%20surface%20codes%20for%20highly%20biased%20noise&amp;journal=Phys.%20Rev.%20X&amp;volume=9&amp;publication_year=2019&amp;author=Tuckett%2CDK\" 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=\"188.\">\n<p class=\"c-article-references__text\" id=\"ref-CR188\">Xu, Q. et al. Tailored xzzx codes for biased noise. Phys. Rev. Res. <b>5<\/b>, 013035 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 188\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tailored%20xzzx%20codes%20for%20biased%20noise&amp;journal=Phys.%20Rev.%20Res.&amp;volume=5&amp;publication_year=2023&amp;author=Xu%2CQ\" 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=\"189.\">\n<p class=\"c-article-references__text\" id=\"ref-CR189\">Tuckett, D. K., Bartlett, S. D. &amp; Flammia, S. T. Ultrahigh error threshold for surface codes with biased noise. Phys. Rev. Lett. <b>120<\/b>, 050505 (2018).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2018PhRvL.120e0505T\" aria-label=\"ADS reference 189\" 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 189\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Ultrahigh%20error%20threshold%20for%20surface%20codes%20with%20biased%20noise&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=120&amp;publication_year=2018&amp;author=Tuckett%2CDK&amp;author=Bartlett%2CSD&amp;author=Flammia%2CST\" 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=\"190.\">\n<p class=\"c-article-references__text\" id=\"ref-CR190\">Bonilla Ataides, J. P., Tuckett, D. K., Bartlett, S. D., Flammia, S. T. &amp; Brown, B. J. The XZZX surface code. Nat. Commun. <b>12<\/b>, 2172 (2021).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021NatCo..12.2172B\" aria-label=\"ADS reference 190\" 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 190\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=The%20XZZX%20surface%20code&amp;journal=Nat.%20Commun.&amp;volume=12&amp;publication_year=2021&amp;author=Bonilla%20Ataides%2CJP&amp;author=Tuckett%2CDK&amp;author=Bartlett%2CSD&amp;author=Flammia%2CST&amp;author=Brown%2CBJ\" 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=\"191.\">\n<p class=\"c-article-references__text\" id=\"ref-CR191\">Dua, A., Kubica, A., Jiang, L., Flammia, S. T. &amp; Gullans, M. J. Clifford-deformed surface codes. PRX Quantum <b>5<\/b>, 010347 (2024).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024PRXQ....5a0347D\" aria-label=\"ADS reference 191\" 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 191\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Clifford-deformed%20surface%20codes&amp;journal=PRX%20Quantum&amp;volume=5&amp;publication_year=2024&amp;author=Dua%2CA&amp;author=Kubica%2CA&amp;author=Jiang%2CL&amp;author=Flammia%2CST&amp;author=Gullans%2CMJ\" 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=\"192.\">\n<p class=\"c-article-references__text\" id=\"ref-CR192\">Kobayashi, F. et al. Tensor-network decoders for process tensor descriptions of non-Markovian noise. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2412.13739\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2412.13739\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2412.13739<\/a> (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"193.\">\n<p class=\"c-article-references__text\" id=\"ref-CR193\">Battistel, F. et al. Real-time decoding for fault-tolerant quantum computing: progress, challenges and outlook. Nano Futures <b>7<\/b>, 032003 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023NanoF...7c2003B\" aria-label=\"ADS reference 193\" 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 193\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Real-time%20decoding%20for%20fault-tolerant%20quantum%20computing%3A%20progress%2C%20challenges%20and%20outlook&amp;journal=Nano%20Futures&amp;volume=7&amp;publication_year=2023&amp;author=Battistel%2CF\" 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=\"194.\">\n<p class=\"c-article-references__text\" id=\"ref-CR194\">Temme, K., Bravyi, S. &amp; Gambetta, J. M. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett. <b>119<\/b>, 180509 (2017).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2017PhRvL.119r0509T\" aria-label=\"ADS reference 194\" 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=3730724\" aria-label=\"MathSciNet reference 194\" 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 194\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Error%20mitigation%20for%20short-depth%20quantum%20circuits&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=119&amp;publication_year=2017&amp;author=Temme%2CK&amp;author=Bravyi%2CS&amp;author=Gambetta%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=\"195.\">\n<p class=\"c-article-references__text\" id=\"ref-CR195\">Li, Y. &amp; Benjamin, S. C. Efficient variational quantum simulator incorporating active error minimization. Phys. Rev. X <b>7<\/b>, 021050 (2017).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 195\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Efficient%20variational%20quantum%20simulator%20incorporating%20active%20error%20minimization&amp;journal=Phys.%20Rev.%20X&amp;volume=7&amp;publication_year=2017&amp;author=Li%2CY&amp;author=Benjamin%2CSC\" 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=\"196.\">\n<p class=\"c-article-references__text\" id=\"ref-CR196\">Guo, Y. &amp; Yang, S. Quantum error mitigation via matrix product operators. PRX Quantum <b>3<\/b>, 040313 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022PRXQ....3d0313G\" aria-label=\"ADS reference 196\" 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 196\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20error%20mitigation%20via%20matrix%20product%20operators&amp;journal=PRX%20Quantum&amp;volume=3&amp;publication_year=2022&amp;author=Guo%2CY&amp;author=Yang%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=\"197.\">\n<p class=\"c-article-references__text\" id=\"ref-CR197\">Tepaske, M. S. &amp; Luitz, D. J. Compressed quantum error mitigation. Phys. Rev. B <b>107<\/b>, L201114 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023PhRvB.107t1114T\" aria-label=\"ADS reference 197\" 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 197\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Compressed%20quantum%20error%20mitigation&amp;journal=Phys.%20Rev.%20B&amp;volume=107&amp;publication_year=2023&amp;author=Tepaske%2CMS&amp;author=Luitz%2CDJ\" 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=\"198.\">\n<p class=\"c-article-references__text\" id=\"ref-CR198\">Filippov, S., Leahy, M., Rossi, M. A. &amp; Garc\u00eda-P\u00e9rez, G. Scalable tensor-network error mitigation for near-term quantum computing. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2307.11740\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2307.11740\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2307.11740<\/a> (2023).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"199.\">\n<p class=\"c-article-references__text\" id=\"ref-CR199\">Filippov, S. N., Maniscalco, S. &amp; Garc\u00eda-P\u00e9rez, G. Scalability of quantum error mitigation techniques: from utility to advantage. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2403.13542\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2403.13542\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2403.13542<\/a> (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"200.\">\n<p class=\"c-article-references__text\" id=\"ref-CR200\">Fischer, L. E. et al. Dynamical simulations of many-body quantum chaos on a quantum computer. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2411.00765\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2411.00765\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2411.00765<\/a> (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"201.\">\n<p class=\"c-article-references__text\" id=\"ref-CR201\">Piveteau, C., Sutter, D., Bravyi, S., Gambetta, J. M. &amp; Temme, K. Error mitigation for universal gates on encoded qubits. Phys. Rev. Lett. <b>127<\/b>, 200505 (2021).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021PhRvL.127t0505P\" aria-label=\"ADS reference 201\" 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=4344500\" aria-label=\"MathSciNet reference 201\" 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 201\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Error%20mitigation%20for%20universal%20gates%20on%20encoded%20qubits&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=127&amp;publication_year=2021&amp;author=Piveteau%2CC&amp;author=Sutter%2CD&amp;author=Bravyi%2CS&amp;author=Gambetta%2CJM&amp;author=Temme%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=\"202.\">\n<p class=\"c-article-references__text\" id=\"ref-CR202\">Novikov, A., Podoprikhin, D., Osokin, A. &amp; Vetrov, D. P. Tensorizing neural networks. In Advances in Neural Information Processing Systems (eds Cortes, C. et al.) Vol. 28, 442\u2013450 (Curran, 2015).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"203.\">\n<p class=\"c-article-references__text\" id=\"ref-CR203\">Novikov, A., Trofimov, M. &amp; Oseledets, I. Exponential machines. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/1605.03795\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/1605.03795\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/1605.03795<\/a> (2016).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"204.\">\n<p class=\"c-article-references__text\" id=\"ref-CR204\">Stoudenmire, E. &amp; Schwab, D. J. Supervised learning with tensor networks. In 30th Conference on Neural Information Processing Systems (NIPS, 2016)<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"205.\">\n<p class=\"c-article-references__text\" id=\"ref-CR205\">Chen, J., Cheng, S., Xie, H., Wang, L. &amp; Xiang, T. Equivalence of restricted Boltzmann machines and tensor network states. Phys. Rev. B <b>97<\/b>, 085104 (2018).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2018PhRvB..97h5104C\" aria-label=\"ADS reference 205\" 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 205\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Equivalence%20of%20restricted%20Boltzmann%20machines%20and%20tensor%20network%20states&amp;journal=Phys.%20Rev.%20B&amp;volume=97&amp;publication_year=2018&amp;author=Chen%2CJ&amp;author=Cheng%2CS&amp;author=Xie%2CH&amp;author=Wang%2CL&amp;author=Xiang%2CT\" 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=\"206.\">\n<p class=\"c-article-references__text\" id=\"ref-CR206\">Li, S., Pan, F., Zhou, P. &amp; Zhang, P. Boltzmann machines as two-dimensional tensor networks. Phys. Rev. B <b>104<\/b>, 075154 (2021).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021PhRvB.104g5154L\" aria-label=\"ADS reference 206\" 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 206\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Boltzmann%20machines%20as%20two-dimensional%20tensor%20networks&amp;journal=Phys.%20Rev.%20B&amp;volume=104&amp;publication_year=2021&amp;author=Li%2CS&amp;author=Pan%2CF&amp;author=Zhou%2CP&amp;author=Zhang%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=\"207.\">\n<p class=\"c-article-references__text\" id=\"ref-CR207\">Han, Z.-Y., Wang, J., Fan, H., Wang, L. &amp; Zhang, P. Unsupervised generative modeling using matrix product states. Phys. Rev.X <b>8<\/b>, 031012 (2018).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 207\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Unsupervised%20generative%20modeling%20using%20matrix%20product%20states&amp;journal=Phys.%20Rev.X&amp;volume=8&amp;publication_year=2018&amp;author=Han%2CZ-Y&amp;author=Wang%2CJ&amp;author=Fan%2CH&amp;author=Wang%2CL&amp;author=Zhang%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=\"208.\">\n<p class=\"c-article-references__text\" id=\"ref-CR208\">Liu, J., Li, S., Zhang, J. &amp; Zhang, P. Tensor networks for unsupervised machine learning. Phys. Rev.E <b>107<\/b>, L012103 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023PhRvE.107a2103L\" aria-label=\"ADS reference 208\" 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=4549417\" aria-label=\"MathSciNet reference 208\" 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 208\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tensor%20networks%20for%20unsupervised%20machine%20learning&amp;journal=Phys.%20Rev.E&amp;volume=107&amp;publication_year=2023&amp;author=Liu%2CJ&amp;author=Li%2CS&amp;author=Zhang%2CJ&amp;author=Zhang%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=\"209.\">\n<p class=\"c-article-references__text\" id=\"ref-CR209\">Glasser, I., Sweke, R., Pancotti, N., Eisert, J. &amp; Cirac, I. Expressive power of tensor-network factorizations for probabilistic modeling. In Advances in Neural Information Processing Systems Vol. 32 pp. 1498\u20131510 (eds Wallach, H. et al.) (Curran, 2019).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"210.\">\n<p class=\"c-article-references__text\" id=\"ref-CR210\">Cheng, S., Wang, L., Xiang, T. &amp; Zhang, P. Tree tensor networks for generative modeling. Phys. Rev. B <b>99<\/b>, 155131 (2019).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2019PhRvB..99o5131C\" aria-label=\"ADS reference 210\" 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 210\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tree%20tensor%20networks%20for%20generative%20modeling&amp;journal=Phys.%20Rev.%20B&amp;volume=99&amp;publication_year=2019&amp;author=Cheng%2CS&amp;author=Wang%2CL&amp;author=Xiang%2CT&amp;author=Zhang%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=\"211.\">\n<p class=\"c-article-references__text\" id=\"ref-CR211\">Vieijra, T., Vanderstraeten, L. &amp; Verstraete, F. Generative modeling with projected entangled-pair states. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2202.08177\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2202.08177\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2202.08177<\/a> (2022).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"212.\">\n<p class=\"c-article-references__text\" id=\"ref-CR212\">Reyes, J. A. &amp; Stoudenmire, E. M. Multi-scale tensor network architecture for machine learning. Mach. Learn. Sci. Technol. <b>2<\/b>, 035036 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 212\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Multi-scale%20tensor%20network%20architecture%20for%20machine%20learning&amp;journal=Mach.%20Learn.%20Sci.%20Technol.&amp;volume=2&amp;publication_year=2021&amp;author=Reyes%2CJA&amp;author=Stoudenmire%2CEM\" 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=\"213.\">\n<p class=\"c-article-references__text\" id=\"ref-CR213\">Huggins, W., Patil, P., Mitchell, B., Whaley, K. B. &amp; Stoudenmire, E. M. Towards quantum machine learning with tensor networks. Quantum Sci. Technol. <b>4<\/b>, 024001 (2019).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2019QS%26T....4b4001H\" aria-label=\"ADS reference 213\" 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 213\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Towards%20quantum%20machine%20learning%20with%20tensor%20networks&amp;journal=Quantum%20Sci.%20Technol.&amp;volume=4&amp;publication_year=2019&amp;author=Huggins%2CW&amp;author=Patil%2CP&amp;author=Mitchell%2CB&amp;author=Whaley%2CKB&amp;author=Stoudenmire%2CEM\" 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=\"214.\">\n<p class=\"c-article-references__text\" id=\"ref-CR214\">Wall, M. L., Abernathy, M. R. &amp; Quiroz, G. Generative machine learning with tensor networks: benchmarks on near-term quantum computers. Phys. Rev. Res. <b>3<\/b>, 023010 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 214\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Generative%20machine%20learning%20with%20tensor%20networks%3A%20benchmarks%20on%20near-term%20quantum%20computers&amp;journal=Phys.%20Rev.%20Res.&amp;volume=3&amp;publication_year=2021&amp;author=Wall%2CML&amp;author=Abernathy%2CMR&amp;author=Quiroz%2CG\" 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=\"215.\">\n<p class=\"c-article-references__text\" id=\"ref-CR215\">Grant, E. et al. Hierarchical quantum classifiers. npj Quantum Inf. <b>4<\/b>, 65 (2018).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2018npjQI...4...65G\" aria-label=\"ADS reference 215\" 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 215\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Hierarchical%20quantum%20classifiers&amp;journal=npj%20Quantum%20Inf.&amp;volume=4&amp;publication_year=2018&amp;author=Grant%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=\"216.\">\n<p class=\"c-article-references__text\" id=\"ref-CR216\">Lazzarin, M., Galli, D. E. &amp; Prati, E. Multi-class quantum classifiers with tensor network circuits for quantum phase recognition. Phys. Lett. A <b>434<\/b>, 128056 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 216\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Multi-class%20quantum%20classifiers%20with%20tensor%20network%20circuits%20for%20quantum%20phase%20recognition&amp;journal=Phys.%20Lett.%20A&amp;volume=434&amp;publication_year=2022&amp;author=Lazzarin%2CM&amp;author=Galli%2CDE&amp;author=Prati%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=\"217.\">\n<p class=\"c-article-references__text\" id=\"ref-CR217\">Rieser, H.-M., K\u00f6ster, F. &amp; Raulf, A. P. Tensor networks for quantum machine learning. Proc. R. Soc. A <b>479<\/b>, 20230218 (2023).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023RSPSA.47930218R\" aria-label=\"ADS reference 217\" 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=4630020\" aria-label=\"MathSciNet reference 217\" 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 217\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Tensor%20networks%20for%20quantum%20machine%20learning&amp;journal=Proc.%20R.%20Soc.%20A&amp;volume=479&amp;publication_year=2023&amp;author=Rieser%2CH-M&amp;author=K%C3%B6ster%2CF&amp;author=Raulf%2CAP\" 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=\"218.\">\n<p class=\"c-article-references__text\" id=\"ref-CR218\">Zhao, C. &amp; Gao, X.-S. Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus. Quantum <b>5<\/b>, 466 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 218\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Analyzing%20the%20barren%20plateau%20phenomenon%20in%20training%20quantum%20neural%20networks%20with%20the%20ZX-calculus&amp;journal=Quantum&amp;volume=5&amp;publication_year=2021&amp;author=Zhao%2CC&amp;author=Gao%2CX-S\" 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=\"219.\">\n<p class=\"c-article-references__text\" id=\"ref-CR219\">Pesah, A. et al. Absence of barren plateaus in quantum convolutional neural networks. Phys. Rev. X <b>11<\/b>, 041011 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 219\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Absence%20of%20barren%20plateaus%20in%20quantum%20convolutional%20neural%20networks&amp;journal=Phys.%20Rev.%20X&amp;volume=11&amp;publication_year=2021&amp;author=Pesah%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=\"220.\">\n<p class=\"c-article-references__text\" id=\"ref-CR220\">Liao, H., Convy, I., Yang, Z. &amp; Whaley, K. B. Decohering tensor network quantum machine learning models. Quant. Mach. Intell. <b>5<\/b>, 7 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 220\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Decohering%20tensor%20network%20quantum%20machine%20learning%20models&amp;journal=Quant.%20Mach.%20Intell.&amp;volume=5&amp;publication_year=2023&amp;author=Liao%2CH&amp;author=Convy%2CI&amp;author=Yang%2CZ&amp;author=Whaley%2CKB\" 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=\"221.\">\n<p class=\"c-article-references__text\" id=\"ref-CR221\">Cong, I., Choi, S. &amp; Lukin, M. D. Quantum convolutional neural networks. Nat. Phys. <b>15<\/b>, 1273\u20131278 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 221\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20convolutional%20neural%20networks&amp;journal=Nat.%20Phys.&amp;volume=15&amp;pages=1273-1278&amp;publication_year=2019&amp;author=Cong%2CI&amp;author=Choi%2CS&amp;author=Lukin%2CMD\" 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=\"222.\">\n<p class=\"c-article-references__text\" id=\"ref-CR222\">Bermejo, P. et al. Quantum convolutional neural networks are (effectively) classically simulable. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2408.12739\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2408.12739\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2408.12739<\/a> (2024).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"223.\">\n<p class=\"c-article-references__text\" id=\"ref-CR223\">Dilip, R., Liu, Y.-J., Smith, A. &amp; Pollmann, F. Data compression for quantum machine learning. Phys. Rev. Res. <b>4<\/b>, 043007 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 223\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Data%20compression%20for%20quantum%20machine%20learning&amp;journal=Phys.%20Rev.%20Res.&amp;volume=4&amp;publication_year=2022&amp;author=Dilip%2CR&amp;author=Liu%2CY-J&amp;author=Smith%2CA&amp;author=Pollmann%2CF\" 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=\"224.\">\n<p class=\"c-article-references__text\" id=\"ref-CR224\">Iaconis, J. &amp; Johri, S. Tensor network based efficient quantum data loading of images. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2310.05897\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2310.05897\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2310.05897<\/a> (2023).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"225.\">\n<p class=\"c-article-references__text\" id=\"ref-CR225\">Dborin, J., Barratt, F., Wimalaweera, V., Wright, L. &amp; Green, A. G. Matrix product state pre-training for quantum machine learning. Quantum Sci. Technol. <b>7<\/b>, 035014 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022QS%26T....7c5014D\" aria-label=\"ADS reference 225\" 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 225\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Matrix%20product%20state%20pre-training%20for%20quantum%20machine%20learning&amp;journal=Quantum%20Sci.%20Technol.&amp;volume=7&amp;publication_year=2022&amp;author=Dborin%2CJ&amp;author=Barratt%2CF&amp;author=Wimalaweera%2CV&amp;author=Wright%2CL&amp;author=Green%2CAG\" 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=\"226.\">\n<p class=\"c-article-references__text\" id=\"ref-CR226\">Rudolph, M. S. et al. Synergy between quantum circuits and tensor networks: short-cutting the race to practical quantum advantage. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2208.13673\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2208.13673\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2208.13673<\/a> (2022).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"227.\">\n<p class=\"c-article-references__text\" id=\"ref-CR227\">Khan, A., Clark, B. K. &amp; Tubman, N. M. Pre-optimizing variational quantum eigensolvers with tensor networks. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2310.12965\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2310.12965\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2310.12965<\/a> (2023).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"228.\">\n<p class=\"c-article-references__text\" id=\"ref-CR228\">Shin, S., Teo, Y. S. &amp; Jeong, H. Dequantizing quantum machine learning models using tensor networks. Phys. Rev. Res. <b>6<\/b>, 023218 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 228\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Dequantizing%20quantum%20machine%20learning%20models%20using%20tensor%20networks&amp;journal=Phys.%20Rev.%20Res.&amp;volume=6&amp;publication_year=2024&amp;author=Shin%2CS&amp;author=Teo%2CYS&amp;author=Jeong%2CH\" 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=\"229.\">\n<p class=\"c-article-references__text\" id=\"ref-CR229\">de Beaudrap, N., Kissinger, A. &amp; Meichanetzidis, K. Tensor network rewriting strategies for satisfiability and counting. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2004.06455\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2004.06455\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2004.06455<\/a> (2020).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"230.\">\n<p class=\"c-article-references__text\" id=\"ref-CR230\">Rakovszky, T., Von Keyserlingk, C. &amp; Pollmann, F. Dissipation-assisted operator evolution method for capturing hydrodynamic transport. Phys. Rev. B <b>105<\/b>, 075131 (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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022PhRvB.105g5131R\" aria-label=\"ADS reference 230\" 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 230\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Dissipation-assisted%20operator%20evolution%20method%20for%20capturing%20hydrodynamic%20transport&amp;journal=Phys.%20Rev.%20B&amp;volume=105&amp;publication_year=2022&amp;author=Rakovszky%2CT&amp;author=Keyserlingk%2CC&amp;author=Pollmann%2CF\" 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=\"231.\">\n<p class=\"c-article-references__text\" id=\"ref-CR231\">Tarabunga, P. S., Tirrito, E., Ba\u00f1uls, M. C. &amp; Dalmonte, M. Nonstabilizerness via matrix product states in the Pauli basis. Phys. Rev. Lett. <b>133<\/b>, 010601 (2024).<\/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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4782349\" aria-label=\"MathSciNet reference 231\" 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 231\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Nonstabilizerness%20via%20matrix%20product%20states%20in%20the%20Pauli%20basis&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=133&amp;publication_year=2024&amp;author=Tarabunga%2CPS&amp;author=Tirrito%2CE&amp;author=Ba%C3%B1uls%2CMC&amp;author=Dalmonte%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=\"232.\">\n<p class=\"c-article-references__text\" id=\"ref-CR232\">Masot-Llima, S. &amp; Garcia-Saez, A. Stabilizer tensor networks: universal quantum simulator on a basis of stabilizer states. Phys. Rev. Lett. <b>133<\/b>, 230601 (2024).<\/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=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4844281\" aria-label=\"MathSciNet reference 232\" 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 232\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Stabilizer%20tensor%20networks%3A%20universal%20quantum%20simulator%20on%20a%20basis%20of%20stabilizer%20states&amp;journal=Phys.%20Rev.%20Lett.&amp;volume=133&amp;publication_year=2024&amp;author=Masot-Llima%2CS&amp;author=Garcia-Saez%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=\"233.\">\n<p class=\"c-article-references__text\" id=\"ref-CR233\">Breuckmann, N. P. &amp; Eberhardt, J. N. Quantum low-density parity-check codes. PRX Quantum <b>2<\/b>, 040101 (2021).<\/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=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021PRXQ....2d0101B\" aria-label=\"ADS reference 233\" 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 233\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Quantum%20low-density%20parity-check%20codes&amp;journal=PRX%20Quantum&amp;volume=2&amp;publication_year=2021&amp;author=Breuckmann%2CNP&amp;author=Eberhardt%2CJN\" 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=\"234.\">\n<p class=\"c-article-references__text\" id=\"ref-CR234\">Panteleev, P. &amp; Kalachev, G. Asymptotically good quantum and locally testable classical LDPC codes. In STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, 375\u2013388 (ACM, 2022).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"235.\">\n<p class=\"c-article-references__text\" id=\"ref-CR235\">Bausch, J. et al. Learning high-accuracy error decoding for quantum processors. Nature <b>635<\/b>, 834\u2013840 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><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 235\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Learning%20high-accuracy%20error%20decoding%20for%20quantum%20processors&amp;journal=Nature&amp;volume=635&amp;pages=834-840&amp;publication_year=2024&amp;author=Bausch%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=\"236.\">\n<p class=\"c-article-references__text\" id=\"ref-CR236\">Zhou, H. H. et al. NVIDIA and QuEra decode quantum errors with AI. NVIDIA Technical Blog <a href=\"https:\/\/developer.nvidia.com\/blog\/nvidia-and-quera-decode-quantum-errors-with-ai\/\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/developer.nvidia.com\/blog\/nvidia-and-quera-decode-quantum-errors-with-ai\/\" target=\"_blank\" rel=\"noopener\">https:\/\/developer.nvidia.com\/blog\/nvidia-and-quera-decode-quantum-errors-with-ai\/<\/a> (2025).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"237.\">\n<p class=\"c-article-references__text\" id=\"ref-CR237\">Cerezo, M. et al. Does provable absence of barren plateaus imply classical simulability? or, why we need to rethink variational quantum computing. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2312.09121\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2312.09121\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2312.09121<\/a> (2023).<\/p>\n<\/li>\n<li class=\"c-article-references__item js-c-reading-companion-references-item\" data-counter=\"238.\">\n<p class=\"c-article-references__text\" id=\"ref-CR238\">Gil-Fuster, E., Gyurik, C., P\u00e9rez-Salinas, A. &amp; Dunjko, V. On the relation between trainability and dequantization of variational quantum learning models. Preprint at <a href=\"https:\/\/arxiv.org\/abs\/2406.07072\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/arxiv.org\/abs\/2406.07072\" target=\"_blank\" rel=\"noopener\">https:\/\/arxiv.org\/abs\/2406.07072<\/a> (2024).<\/p>\n<\/li>\n","protected":false},"excerpt":{"rendered":"Schollw\u00f6ck, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326, 96\u2013192 (2011).&hellip;\n","protected":false},"author":2,"featured_media":304392,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3164],"tags":[3284,3968,74,7030,15109,53,16,15],"class_list":{"0":"post-304391","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-computing","8":"tag-computing","9":"tag-general","10":"tag-physics","11":"tag-quantum-information","12":"tag-qubits","13":"tag-technology","14":"tag-uk","15":"tag-united-kingdom"},"share_on_mastodon":{"url":"https:\/\/pubeurope.com\/@uk\/114943230144364922","error":""},"_links":{"self":[{"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/posts\/304391","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=304391"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/posts\/304391\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/media\/304392"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/media?parent=304391"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/categories?post=304391"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/uk\/wp-json\/wp\/v2\/tags?post=304391"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}