{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,23]],"date-time":"2025-10-23T11:20:58Z","timestamp":1761218458568},"reference-count":48,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,2,3]],"date-time":"2022-02-03T00:00:00Z","timestamp":1643846400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"German Research Foundation","doi-asserted-by":"crossref","award":["CRC 183, EI 519\/15-1"],"award-info":[{"award-number":["CRC 183, EI 519\/15-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100007601","name":"EU Horizon 2020","doi-asserted-by":"crossref","award":["817482"],"award-info":[{"award-number":["817482"]}],"id":[{"id":"10.13039\/501100007601","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Quantum Information National Laboratory of Hungary","award":["K124176, FK135220, K124351"],"award-info":[{"award-number":["K124176, FK135220, K124351"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>The study of critical quantum many-body systems through conformal field theory (CFT) is one of the pillars of modern quantum physics. Certain CFTs are also understood to be dual to higher-dimensional theories of gravity via the anti-de Sitter\/conformal field theory (AdS\/CFT) correspondence. To reproduce various features of AdS\/CFT, a large number of discrete models based on tensor networks have been proposed. Some recent models, most notably including toy models of holographic quantum error correction, are constructed on regular time-slice discretizations of AdS. In this work, we show that the symmetries of these models are well suited for approximating CFT states, as their geometry enforces a discrete subgroup of conformal symmetries. Based on these symmetries, we introduce the notion of a quasiperiodic conformal field theory (qCFT), a critical theory less restrictive than a full CFT and with characteristic multi-scale quasiperiodicity. We discuss holographic code states and their renormalization group flow as specific implementations of a qCFT with fractional central charges and argue that their behavior generalizes to a large class of existing and future models. Beyond approximating CFT properties, we show that these can be best understood as belonging to a paradigm of discrete holography.<\/jats:p>","DOI":"10.22331\/q-2022-02-03-643","type":"journal-article","created":{"date-parts":[[2022,2,3]],"date-time":"2022-02-03T17:51:34Z","timestamp":1643910694000},"page":"643","update-policy":"http:\/\/dx.doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":19,"title":["Tensor network models of AdS\/qCFT"],"prefix":"10.22331","volume":"6","author":[{"given":"Alexander","family":"Jahn","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, 14195 Berlin, Germany"},{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA"}]},{"given":"Zolt\u00e1n","family":"Zimbor\u00e1s","sequence":"additional","affiliation":[{"name":"Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, 1121 Budapest, Hungary"},{"name":"BME-MTA Lend\u00fclet Quantum Information Theory Research Group, 1111 Budapest, Hungary"},{"name":"Faculty of Informatics, E\u00f6tv\u00f6s Lor\u00e1nd University, 1117 Budapest, Hungary"}]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, 14195 Berlin, Germany"},{"name":"Department of Mathematics and Computer Science, Freie Universit\u00e4t Berlin, 14195 Berlin, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2022,2,3]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"P. Francesco, P. Mathieu, and D. Senechal. Conformal field theory. Springer, Berlin, 1997. 10.1007\/978-1-4612-2256-9.","DOI":"10.1007\/978-1-4612-2256-9"},{"key":"1","unstructured":"P. H. Ginsparg. Applied Conformal Field Theory. In Les Houches Summer School in Theoretical Physics: Fields, Strings, Critical Phenomena, 1988. URL https:\/\/arxiv.org\/abs\/hep-th\/9108028."},{"key":"2","doi-asserted-by":"publisher","unstructured":"R. Blumenhagen and E. Plauschinn. Introduction to conformal field theory: with applications to String theory, volume 779. Springer, 2009. 10.1007\/978-3-642-00450-6.","DOI":"10.1007\/978-3-642-00450-6"},{"key":"3","doi-asserted-by":"publisher","unstructured":"J. M. Maldacena. The Large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys., 38: 1113\u20131133, 1999. 10.1023\/A:1026654312961. [Adv. Theor. Math. Phys. 2, 231(1998)].","DOI":"10.1023\/A:1026654312961"},{"key":"4","doi-asserted-by":"publisher","unstructured":"E. Witten. Anti-de Sitter space and holography. Adv. Theor. Math. Phys., 2: 253\u2013291, 1998. 10.4310\/ATMP.1998.v2.n2.a2.","DOI":"10.4310\/ATMP.1998.v2.n2.a2"},{"key":"5","doi-asserted-by":"publisher","unstructured":"G. Vidal. Class of quantum many-body states that can be efficiently simulated. Phys. Rev. Lett., 101: 110501, 2008. 10.1103\/PhysRevLett.101.110501.","DOI":"10.1103\/PhysRevLett.101.110501"},{"key":"6","doi-asserted-by":"publisher","unstructured":"R. N. C. Pfeifer, G. Evenbly, and G. Vidal. Entanglement renormalization, scale invariance, and quantum criticality. Phys. Rev. A, 79: 040301, 2009. 10.1103\/PhysRevA.79.040301.","DOI":"10.1103\/PhysRevA.79.040301"},{"key":"7","unstructured":"A. Milsted and G. Vidal. Tensor networks as conformal transformations, 2018a. URL https:\/\/arxiv.org\/abs\/1805.12524."},{"key":"8","doi-asserted-by":"publisher","unstructured":"G. Evenbly and G. Vidal. Entanglement renormalization in two spatial dimensions. Phys. Rev. Lett., 102: 180406, 2009. 10.1103\/PhysRevLett.102.180406.","DOI":"10.1103\/PhysRevLett.102.180406"},{"key":"9","doi-asserted-by":"publisher","unstructured":"B. Swingle. Entanglement renormalization and holography. Phys. Rev. D, 86: 065007, 2012. 10.1103\/PhysRevD.86.065007.","DOI":"10.1103\/PhysRevD.86.065007"},{"key":"10","doi-asserted-by":"publisher","unstructured":"S. Singh. Tensor network state correspondence and holography. Phys. Rev. D, 97: 026012, 2018. 10.1103\/PhysRevD.97.026012.","DOI":"10.1103\/PhysRevD.97.026012"},{"key":"11","doi-asserted-by":"publisher","unstructured":"C. Beny. Causal structure of the entanglement renormalization ansatz. New J. Phys., 15: 023020, 2013. 10.1088\/1367-2630\/15\/2\/023020.","DOI":"10.1088\/1367-2630\/15\/2\/023020"},{"key":"12","doi-asserted-by":"publisher","unstructured":"N. Bao, C. J. Cao, S. M. Carroll, A. Chatwin-Davies, N. Hunter-Jones, J. Pollack, and G. N. Remmen. Consistency conditions for an AdS multiscale entanglement renormalization ansatz correspondence. Phys. Rev. D, 91: 125036, 2015. 10.1103\/PhysRevD.91.125036.","DOI":"10.1103\/PhysRevD.91.125036"},{"key":"13","unstructured":"A. Milsted and G. Vidal. Geometric interpretation of the multi-scale entanglement renormalization ansatz, 2018b. URL https:\/\/arxiv.org\/abs\/1812.00529."},{"key":"14","doi-asserted-by":"publisher","unstructured":"A. Jahn and J. Eisert. Holographic tensor network models and quantum error correction: a topical review. Quantum Sci. Technol., 6: 033002, 2021. 10.1088\/2058-9565\/ac0293.","DOI":"10.1088\/2058-9565\/ac0293"},{"key":"15","doi-asserted-by":"publisher","unstructured":"L. Boyle, M. Dickens, and F. Flicker. Conformal quasicrystals and holography. Phys. Rev. X, 10 (1): 011009, 2020. 10.1103\/PhysRevX.10.011009.","DOI":"10.1103\/PhysRevX.10.011009"},{"key":"16","doi-asserted-by":"publisher","unstructured":"A. Jahn, M. Gluza, F. Pastawski, and J. Eisert. Majorana dimers and holographic quantum error-correcting codes. Phys. Rev. Research, 1: 033079, 2019a. 10.1103\/PhysRevResearch.1.033079.","DOI":"10.1103\/PhysRevResearch.1.033079"},{"key":"17","doi-asserted-by":"publisher","unstructured":"S. Ryu and T. Takayanagi. Holographic derivation of entanglement entropy from the anti-de Sitter space\/conformal field theory correspondence. Phys. Rev. Lett., 96: 181602, 2006. 10.1103\/PhysRevLett.96.181602.","DOI":"10.1103\/PhysRevLett.96.181602"},{"key":"18","doi-asserted-by":"publisher","unstructured":"H. Casini, M. Huerta, and R. C. Myers. Towards a derivation of holographic entanglement entropy. JHEP, 05: 036, 2011. 10.1007\/JHEP05(2011)036.","DOI":"10.1007\/JHEP05(2011)036"},{"key":"19","doi-asserted-by":"publisher","unstructured":"A. Bhattacharyya, Z.-S. Gao, L.-Y. Hung, and S.-N. Liu. Exploring the tensor networks\/AdS correspondence. JHEP, 08: 086, 2016. 10.1007\/JHEP08(2016)086.","DOI":"10.1007\/JHEP08(2016)086"},{"key":"20","doi-asserted-by":"publisher","unstructured":"T. J. Osborne and D. E. Stiegemann. Dynamics for holographic codes. JHEP, 04: 154, 2020. 10.1007\/JHEP04(2020)154.","DOI":"10.1007\/JHEP04(2020)154"},{"key":"21","doi-asserted-by":"publisher","unstructured":"T. Kohler and T. Cubitt. Toy models of holographic duality between local Hamiltonians. JHEP, 08: 017, 2019. 10.1007\/JHEP08(2019)017.","DOI":"10.1007\/JHEP08(2019)017"},{"key":"22","doi-asserted-by":"publisher","unstructured":"J. Maciejko and S. Rayan. Hyperbolic band theory. Sci. Adv., 7: abe9170, 2021. 10.1126\/sciadv.abe9170.","DOI":"10.1126\/sciadv.abe9170"},{"key":"23","doi-asserted-by":"crossref","unstructured":"I. Boettcher, A. V. Gorshkov, A. J. Koll\u00e1r, J. Maciejko, S. Rayan, and R. Thomale. Crystallography of hyperbolic lattices, 2021. URL https:\/\/arxiv.org\/abs\/2105.01087.","DOI":"10.1103\/PhysRevB.105.125118"},{"key":"24","doi-asserted-by":"publisher","unstructured":"G. Evenbly. Hyperinvariant tensor networks and holography. Phys. Rev. Lett., 119: 141602, 2017. 10.1103\/PhysRevLett.119.141602.","DOI":"10.1103\/PhysRevLett.119.141602"},{"key":"25","doi-asserted-by":"publisher","unstructured":"A. Jahn, Z. Zimbor\u00e1s, and J. Eisert. Central charges of aperiodic holographic tensor network models. Phys. Rev. A, 102: 042407, 2020. 10.1103\/PhysRevA.102.042407.","DOI":"10.1103\/PhysRevA.102.042407"},{"key":"26","doi-asserted-by":"publisher","unstructured":"A. Jahn, M. Gluza, F. Pastawski, and J. Eisert. Holography and criticality in matchgate tensor networks. Sci. Adv., 5: eaaw0092, 2019b. 10.1126\/sciadv.aaw0092.","DOI":"10.1126\/sciadv.aaw0092"},{"key":"27","doi-asserted-by":"publisher","unstructured":"C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters. Mixed state entanglement and quantum error correction. Phys. Rev. A, 54: 3824\u20133851, 1996. 10.1103\/PhysRevA.54.3824.","DOI":"10.1103\/PhysRevA.54.3824"},{"key":"28","doi-asserted-by":"publisher","unstructured":"R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek. Perfect quantum error correcting code. Phys. Rev. Lett., 77: 198\u2013201, 1996. 10.1103\/PhysRevLett.77.198.","DOI":"10.1103\/PhysRevLett.77.198"},{"key":"29","doi-asserted-by":"publisher","unstructured":"F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill. Holographic quantum error-correcting codes: Toy models for the bulk\/boundary correspondence. JHEP, 2015: 149, 2015. 10.1007\/JHEP06(2015)149.","DOI":"10.1007\/JHEP06(2015)149"},{"key":"30","doi-asserted-by":"publisher","unstructured":"J. Eisert, M. Cramer, and M. B. Plenio. Area laws for the entanglement entropy. Rev. Mod. Phys., 82: 277\u2013306, 2010. 10.1103\/RevModPhys.82.277.","DOI":"10.1103\/RevModPhys.82.277"},{"key":"31","doi-asserted-by":"publisher","unstructured":"P. Calabrese and J. L. Cardy. Entanglement entropy and quantum field theory. J. Stat. Mech., 0406: P06002, 2004. 10.1088\/1742-5468\/2004\/06\/P06002.","DOI":"10.1088\/1742-5468\/2004\/06\/P06002"},{"key":"32","doi-asserted-by":"publisher","unstructured":"J. D. Brown and M. Henneaux. Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity. Commun. Math. Phys., 104: 207\u2013226, 1986. 10.1007\/BF01211590.","DOI":"10.1007\/BF01211590"},{"key":"33","doi-asserted-by":"publisher","unstructured":"G. Vidal. Entanglement renormalization: An introduction. Understanding quantum phase transitions, 2010. 10.1201\/b10273. URL https:\/\/arxiv.org\/abs\/0912.1651.","DOI":"10.1201\/b10273"},{"key":"34","doi-asserted-by":"publisher","unstructured":"B. Czech, G. Evenbly, L. Lamprou, S. McCandlish, X.-L. Qi, J. Sully, and G. Vidal. Tensor network quotient takes the vacuum to the thermal state. Phys. Rev. B, 94: 085101, 2016. 10.1103\/PhysRevB.94.085101.","DOI":"10.1103\/PhysRevB.94.085101"},{"key":"35","doi-asserted-by":"publisher","unstructured":"R. Juh\u00e1sz and Z. Zimbor\u00e1s. Entanglement entropy in aperiodic singlet phases. J. Stat. Mech., 2007 (4): 04004, 2007. 10.1088\/1742-5468\/2007\/04\/P04004.","DOI":"10.1088\/1742-5468\/2007\/04\/P04004"},{"key":"36","doi-asserted-by":"publisher","unstructured":"F. Igl\u00f3i and C. Monthus. Strong disorder RG approach - a short review of recent developments. Eur. Phys. J. B, 91: 290, 2018. 10.1140\/epjb\/e2018-90434-8.","DOI":"10.1140\/epjb\/e2018-90434-8"},{"key":"37","doi-asserted-by":"publisher","unstructured":"M. Steinberg and J. Prior. Conformal properties of hyperinvariant tensor networks. Sci. Rep., 12: 532, 2022. 10.1038\/s41598-021-04375-5.","DOI":"10.1038\/s41598-021-04375-5"},{"key":"38","doi-asserted-by":"publisher","unstructured":"R. J. Harris, N. A. McMahon, G. K. Brennen, and T. M. Stace. Calderbank-Shor-Steane holographic quantum error-correcting codes. Phys. Rev. A, 98: 052301, 2018. 10.1103\/PhysRevA.98.052301.","DOI":"10.1103\/PhysRevA.98.052301"},{"key":"39","doi-asserted-by":"publisher","unstructured":"P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter, and Z. Yang. Holographic duality from random tensor networks. JHEP, 11: 009, 2016. 10.1007\/JHEP11(2016)009.","DOI":"10.1007\/JHEP11(2016)009"},{"key":"40","doi-asserted-by":"publisher","unstructured":"S. S. Gubser, J. Knaute, S. Parikh, A. Samberg, and P. Witaszczyk. $p$-adic AdS\/CFT. Commun. Math. Phys., 352: 1019\u20131059, 2017. 10.1007\/s00220-016-2813-6.","DOI":"10.1007\/s00220-016-2813-6"},{"key":"41","doi-asserted-by":"publisher","unstructured":"M. Heydeman, M. Marcolli, I. Saberi, and B. Stoica. Tensor networks, $p$-adic fields, and algebraic curves: Arithmetic and the AdS$_3$\/CFT$_2$ correspondence. Adv. Theor. Math. Phys., 22: 93\u2013176, 2018. 10.4310\/ATMP.2018.v22.n1.a4.","DOI":"10.4310\/ATMP.2018.v22.n1.a4"},{"key":"42","doi-asserted-by":"publisher","unstructured":"D. S. Fisher. Critical behavior of random transverse-field ising spin chains. Phys. Rev. B, 51: 6411\u20136461, 1995. 10.1103\/PhysRevB.51.6411.","DOI":"10.1103\/PhysRevB.51.6411"},{"key":"43","doi-asserted-by":"publisher","unstructured":"G. Refael and J. E. Moore. Entanglement entropy of random quantum critical points in one dimension. Phys. Rev. Lett., 93: 260602, 2004. 10.1103\/PhysRevLett.93.260602.","DOI":"10.1103\/PhysRevLett.93.260602"},{"key":"44","doi-asserted-by":"publisher","unstructured":"R. Vosk, D. A. Huse, and E. Altman. Theory of the many-body localization transition in one-dimensional systems. Phys. Rev. X, 5: 031032, 2015. 10.1103\/PhysRevX.5.031032.","DOI":"10.1103\/PhysRevX.5.031032"},{"key":"45","doi-asserted-by":"publisher","unstructured":"Z.-L. Tsai, P. Chen, and Y.-C. Lin. Tensor network renormalization group study of spin-1 random Heisenberg chains. Europ. Phys. J. B, 93, 2020. 10.1140\/epjb\/e2020-100585-8.","DOI":"10.1140\/epjb\/e2020-100585-8"},{"key":"46","doi-asserted-by":"publisher","unstructured":"I. V. Protopopov, R. K. Panda, T. Parolini, A. Scardicchio, E. Demler, and D. A. Abanin. Non-abelian symmetries and disorder: A broad nonergodic regime and anomalous thermalization. Phys. Rev. X, 10: 011025, 2020. 10.1103\/PhysRevX.10.011025.","DOI":"10.1103\/PhysRevX.10.011025"},{"key":"47","doi-asserted-by":"publisher","unstructured":"I. H. Kim and M. J. Kastoryano. Entanglement renormalization, quantum error correction, and bulk causality. JHEP, 2017: 40, 2017. 10.1007\/JHEP04(2017)040.","DOI":"10.1007\/JHEP04(2017)040"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2022-02-03-643\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,1,25]],"date-time":"2023-01-25T19:50:42Z","timestamp":1674676242000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2022-02-03-643\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,3]]},"references-count":48,"URL":"https:\/\/doi.org\/10.22331\/q-2022-02-03-643","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,3]]},"article-number":"643"}}