# Tensor network models of AdS/qCFT

Alexander Jahn1,2, Zoltán Zimborás3,4,5, and Jens Eisert1,6

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

### 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.

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.

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[2] 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).

[3] 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).

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

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

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

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

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

[9] Matthew Steinberg and Javier Prior, "Conformal Properties of Hyperinvariant Tensor Networks", arXiv:2012.09591, Scientific Reports 12, 532 (2020).

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

[11] 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.

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

[13] Pablo Basteiro, Giuseppe Di Giulio, Johanna Erdmenger, Jonathan Karl, René Meyer, and Zhuo-Yu Xian, "Towards Explicit Discrete Holography: Aperiodic Spin Chains from Hyperbolic Tilings", arXiv:2205.05693.

[14] 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).

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