Tensor network models of AdS/qCFT

Alexander Jahn; Zoltán Zimborás; Jens Eisert

doi:10.22331/q-2022-02-03-643

Tensor network models of AdS/qCFT

Alexander Jahn^1,2, Zoltán Zimborás^3,4,5, and Jens Eisert^1,6

¹Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
²Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
³Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, 1121 Budapest, Hungary
⁴BME-MTA Lendület Quantum Information Theory Research Group, 1111 Budapest, Hungary
⁵Faculty of Informatics, Eötvös Loránd University, 1117 Budapest, Hungary
⁶Department of Mathematics and Computer Science, Freie Universität Berlin, 14195 Berlin, Germany

Published:	2022-02-03, volume 6, page 643
Eprint:	arXiv:2004.04173v4
Doi:	https://doi.org/10.22331/q-2022-02-03-643
Citation:	Quantum 6, 643 (2022).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

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.

Featured image: How to embed two tensor networks into 2+1-dimensional anti-de Sitter (AdS) spacetime: The multi-scale entanglement renormalization ansatz (MERA, left figure) can be identified with the symmetries of a discretized AdS lightcone, whereas regular tilings naturally discretize slices at constant time (right figure). The AdS coordinates $\rho$ (radial scale) and $t$ (time) are shown explicitly, with another angular coordinate omitted for simplicity.

Popular summary

Progress on the study of holographic dualities has revealed deep connections between conformal field theory (CFT), which describes the scale-independent physics occuring at quantum critical points, and the geometry of hyperbolic anti-de Sitter (AdS) spacetimes. In particular, the same continuous symmetries appear in both settings. In discrete lattice models such as tensor networks, however, the bulk symmetries of the hyperbolic geometry are broken. We show that for highly symmetric regular lattices, these discrete symmetries lead to multi-scale quasiperiodicity on the boundary, which still encodes a discrete subset of CFT symmetries. We call theories respecting these symmetries quasiperiodic conformal field theories (qCFTs) and show how far they resemble the more constrained CFTs. We also show that previously constructed tensor network models already realize properties of our proposal.

► BibTeX data

@article{Jahn2022tensornetworkmodels,
  doi = {10.22331/q-2022-02-03-643},
  url = {https://doi.org/10.22331/q-2022-02-03-643},
  title = {Tensor network models of {A}d{S}/q{CFT}},
  author = {Jahn, Alexander and Zimbor{\'{a}}s, Zolt{\'{a}}n and Eisert, Jens},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {643},
  month = feb,
  year = {2022}
}

► References

[1] P. Francesco, P. Mathieu, and D. Senechal. Conformal field theory. Springer, Berlin, 1997. 10.1007/978-1-4612-2256-9.
https://doi.org/10.1007/978-1-4612-2256-9

[2] 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.
https://arxiv.org/abs/hep-th/9108028

[3] 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.
https://doi.org/10.1007/978-3-642-00450-6

[4] J. M. Maldacena. The Large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys., 38: 1113–1133, 1999. 10.1023/A:1026654312961. [Adv. Theor. Math. Phys. 2, 231(1998)].
https://doi.org/10.1023/A:1026654312961

[5] E. Witten. Anti-de Sitter space and holography. Adv. Theor. Math. Phys., 2: 253–291, 1998. 10.4310/ATMP.1998.v2.n2.a2.
https://doi.org/10.4310/ATMP.1998.v2.n2.a2

[6] G. Vidal. Class of quantum many-body states that can be efficiently simulated. Phys. Rev. Lett., 101: 110501, 2008. 10.1103/PhysRevLett.101.110501.
https://doi.org/10.1103/PhysRevLett.101.110501

[7] 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.
https://doi.org/10.1103/PhysRevA.79.040301

[8] A. Milsted and G. Vidal. Tensor networks as conformal transformations, 2018a. URL https://arxiv.org/abs/1805.12524.
arXiv:1805.12524

[9] G. Evenbly and G. Vidal. Entanglement renormalization in two spatial dimensions. Phys. Rev. Lett., 102: 180406, 2009. 10.1103/PhysRevLett.102.180406.
https://doi.org/10.1103/PhysRevLett.102.180406

[10] B. Swingle. Entanglement renormalization and holography. Phys. Rev. D, 86: 065007, 2012. 10.1103/PhysRevD.86.065007.
https://doi.org/10.1103/PhysRevD.86.065007

[11] S. Singh. Tensor network state correspondence and holography. Phys. Rev. D, 97: 026012, 2018. 10.1103/PhysRevD.97.026012.
https://doi.org/10.1103/PhysRevD.97.026012

[12] C. Beny. Causal structure of the entanglement renormalization ansatz. New J. Phys., 15: 023020, 2013. 10.1088/1367-2630/15/2/023020.
https://doi.org/10.1088/1367-2630/15/2/023020

[13] 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.
https://doi.org/10.1103/PhysRevD.91.125036

[14] A. Milsted and G. Vidal. Geometric interpretation of the multi-scale entanglement renormalization ansatz, 2018b. URL https://arxiv.org/abs/1812.00529.
arXiv:1812.00529

[15] 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.
https://doi.org/10.1088/2058-9565/ac0293

[16] L. Boyle, M. Dickens, and F. Flicker. Conformal quasicrystals and holography. Phys. Rev. X, 10 (1): 011009, 2020. 10.1103/PhysRevX.10.011009.
https://doi.org/10.1103/PhysRevX.10.011009

[17] 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.
https://doi.org/10.1103/PhysRevResearch.1.033079

[18] 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.
https://doi.org/10.1103/PhysRevLett.96.181602

[19] H. Casini, M. Huerta, and R. C. Myers. Towards a derivation of holographic entanglement entropy. JHEP, 05: 036, 2011. 10.1007/JHEP05(2011)036.
https://doi.org/10.1007/JHEP05(2011)036

[20] 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.
https://doi.org/10.1007/JHEP08(2016)086

[21] T. J. Osborne and D. E. Stiegemann. Dynamics for holographic codes. JHEP, 04: 154, 2020. 10.1007/JHEP04(2020)154.
https://doi.org/10.1007/JHEP04(2020)154

[22] T. Kohler and T. Cubitt. Toy models of holographic duality between local Hamiltonians. JHEP, 08: 017, 2019. 10.1007/JHEP08(2019)017.
https://doi.org/10.1007/JHEP08(2019)017

[23] J. Maciejko and S. Rayan. Hyperbolic band theory. Sci. Adv., 7: abe9170, 2021. 10.1126/sciadv.abe9170.
https://doi.org/10.1126/sciadv.abe9170

[24] I. Boettcher, A. V. Gorshkov, A. J. Kollár, J. Maciejko, S. Rayan, and R. Thomale. Crystallography of hyperbolic lattices, 2021. URL https://arxiv.org/abs/2105.01087.
arXiv:2105.01087

[25] G. Evenbly. Hyperinvariant tensor networks and holography. Phys. Rev. Lett., 119: 141602, 2017. 10.1103/PhysRevLett.119.141602.
https://doi.org/10.1103/PhysRevLett.119.141602

[26] A. Jahn, Z. Zimborás, and J. Eisert. Central charges of aperiodic holographic tensor network models. Phys. Rev. A, 102: 042407, 2020. 10.1103/PhysRevA.102.042407.
https://doi.org/10.1103/PhysRevA.102.042407

[27] 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.
https://doi.org/10.1126/sciadv.aaw0092

[28] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters. Mixed state entanglement and quantum error correction. Phys. Rev. A, 54: 3824–3851, 1996. 10.1103/PhysRevA.54.3824.
https://doi.org/10.1103/PhysRevA.54.3824

[29] R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek. Perfect quantum error correcting code. Phys. Rev. Lett., 77: 198–201, 1996. 10.1103/PhysRevLett.77.198.
https://doi.org/10.1103/PhysRevLett.77.198

[30] 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.
https://doi.org/10.1007/JHEP06(2015)149

[31] J. Eisert, M. Cramer, and M. B. Plenio. Area laws for the entanglement entropy. Rev. Mod. Phys., 82: 277–306, 2010. 10.1103/RevModPhys.82.277.
https://doi.org/10.1103/RevModPhys.82.277

[32] 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.
https://doi.org/10.1088/1742-5468/2004/06/P06002

[33] 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: 207–226, 1986. 10.1007/BF01211590.
https://doi.org/10.1007/BF01211590

[34] G. Vidal. Entanglement renormalization: An introduction. Understanding quantum phase transitions, 2010. 10.1201/b10273. URL https://arxiv.org/abs/0912.1651.
https://doi.org/10.1201/b10273
arXiv:0912.1651

[35] 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.
https://doi.org/10.1103/PhysRevB.94.085101

[36] R. Juhász and Z. Zimborás. Entanglement entropy in aperiodic singlet phases. J. Stat. Mech., 2007 (4): 04004, 2007. 10.1088/1742-5468/2007/04/P04004.
https://doi.org/10.1088/1742-5468/2007/04/P04004

[37] F. Iglói 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.
https://doi.org/10.1140/epjb/e2018-90434-8

[38] M. Steinberg and J. Prior. Conformal properties of hyperinvariant tensor networks. Sci. Rep., 12: 532, 2022. 10.1038/s41598-021-04375-5.
https://doi.org/10.1038/s41598-021-04375-5

[39] 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.
https://doi.org/10.1103/PhysRevA.98.052301

[40] 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.
https://doi.org/10.1007/JHEP11(2016)009

[41] S. S. Gubser, J. Knaute, S. Parikh, A. Samberg, and P. Witaszczyk. $p$-adic AdS/CFT. Commun. Math. Phys., 352: 1019–1059, 2017. 10.1007/s00220-016-2813-6.
https://doi.org/10.1007/s00220-016-2813-6

[42] 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: 93–176, 2018. 10.4310/ATMP.2018.v22.n1.a4.
https://doi.org/10.4310/ATMP.2018.v22.n1.a4

[43] D. S. Fisher. Critical behavior of random transverse-field ising spin chains. Phys. Rev. B, 51: 6411–6461, 1995. 10.1103/PhysRevB.51.6411.
https://doi.org/10.1103/PhysRevB.51.6411

[44] 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.
https://doi.org/10.1103/PhysRevLett.93.260602

[45] 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.
https://doi.org/10.1103/PhysRevX.5.031032

[46] 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.
https://doi.org/10.1140/epjb/e2020-100585-8

[47] 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.
https://doi.org/10.1103/PhysRevX.10.011025

[48] I. H. Kim and M. J. Kastoryano. Entanglement renormalization, quantum error correction, and bulk causality. JHEP, 2017: 40, 2017. 10.1007/JHEP04(2017)040.
https://doi.org/10.1007/JHEP04(2017)040

Cited by

[1] Valérie Bettaque and Brian Swingle, "NoRA: A Tensor Network Ansatz for Volume-Law Entangled Equilibrium States of Highly Connected Hamiltonians", Quantum 8, 1362 (2024).

[2] Johannes Nokkala, Jyrki Piilo, and Ginestra Bianconi, "Complex quantum networks: a topical review", Journal of Physics A: Mathematical and Theoretical 57 23, 233001 (2024).

[3] Jason Pollack, Patrick Rall, and Andrea Rocchetto, "Understanding holographic error correction via unique algebras and atomic examples", Journal of High Energy Physics 2022 6, 56 (2022).

[4] Takato Mori, Hidetaka Manabe, and Hiroaki Matsueda, "Entanglement distillation toward minimal bond cut surface in tensor networks", Physical Review D 106 8, 086008 (2022).

[5] Terry Farrelly, Nicholas Milicevic, Robert J. Harris, Nathan A. McMahon, and Thomas M. Stace, "Parallel decoding of multiple logical qubits in tensor-network codes", Physical Review A 105 5, 052446 (2022).

[6] Matthew Steinberg, Sebastian Feld, and Alexander Jahn, "Holographic codes from hyperinvariant tensor networks", Nature Communications 14 1, 7314 (2023).

[7] A. Jahn, M. Gluza, C. Verhoeven, S. Singh, and J. Eisert, "Boundary theories of critical matchgate tensor networks", Journal of High Energy Physics 2022 4, 111 (2022).

[8] ChunJun Cao and Brad Lackey, "Quantum Weight Enumerators and Tensor Networks", IEEE Transactions on Information Theory 70 5, 3512 (2024).

[9] Pablo Basteiro, Giuseppe Di Giulio, Johanna Erdmenger, Jonathan Karl, Rene Meyer, and Zhuo-Yu Xian, "Towards explicit discrete holography: Aperiodic spin chains from hyperbolic tilings", SciPost Physics 13 5, 103 (2022).

[10] Muhammad Asaduzzaman, Simon Catterall, Yannick Meurice, and Goksu Can Toga, "Quantum Ising model on two-dimensional anti–de Sitter space", Physical Review D 109 5, 054513 (2024).

[11] Alexander Jahn and Jens Eisert, "Holographic tensor network models and quantum error correction: a topical review", Quantum Science and Technology 6 3, 033002 (2021).

[12] Christopher David White, ChunJun Cao, and Brian Swingle, "Conformal field theories are magical", Physical Review B 103 7, 075145 (2021).

[13] ChunJun Cao and Brad Lackey, "Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks", PRX Quantum 3 2, 020332 (2022).

[14] Jan Boruch, Pawel Caputa, Dongsheng Ge, and Tadashi Takayanagi, "Holographic path-integral optimization", Journal of High Energy Physics 2021 7, 16 (2021).

[15] Johanna Erdmenger, Jonathan Karl, Yanick Thurn, Matthias Vojta, and Zhuo-Yu Xian, "Discrete JT gravity as an Ising model", arXiv:2407.06266, (2024).

[16] ChunJun Cao and Brad Lackey, "Approximate Bacon-Shor code and holography", Journal of High Energy Physics 2021 5, 127 (2021).

[17] Jan Boruch, Pawel Caputa, and Tadashi Takayanagi, "Path integral optimization from Hartle-Hawking wave function", Physical Review D 103 4, 046017 (2021).

[18] Matthew Steinberg and Javier Prior, "Conformal properties of hyperinvariant tensor networks", Scientific Reports 12, 532 (2022).

[19] Takato Mori, "Entanglement structure in quantum many-body systems, field theories, and holography", arXiv:2308.09758, (2023).

[20] ChunJun Cao, Jason Pollack, and Yixu Wang, "Hyper-Invariant MERA: Approximate Holographic Error Correction Codes with Power-Law Correlations", arXiv:2103.08631, (2021).

[21] Susan Hao, Piyush Kumar, Sarah Laszlo, Shivani Poddar, Bhaktipriya Radharapu, and Renee Shelby, "Safety and Fairness for Content Moderation in Generative Models", arXiv:2306.06135, (2023).

[22] Terry Farrelly, Robert J. Harris, Nathan A. McMahon, and Thomas M. Stace, "Parallel decoding of multiple logical qubits in tensor-network codes", arXiv:2012.07317, (2020).

[23] ChunJun Cao, Jason Pollack, and Yixu Wang, "Hyperinvariant multiscale entanglement renormalization ansatz: Approximate holographic error correction codes with power-law correlations", Physical Review D 105 2, 026018 (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2024-09-16 13:09:58) and SAO/NASA ADS (last updated successfully 2024-09-16 13:09:59). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.