Holographic tensor-network code[14] 


Quantum Lego code whose encoding isometry forms a holographic tensor network, i.e., a tensor network associated with a tiling of hyperbolic space. Physical qubits are associated with uncontracted tensor legs at the boundary of the tesselation, while logical qubits are associated with uncontracted legs in the bulk. The number of layers emanating form the central point of the tiling is the radius of the code.

The encoding map models radial time evolution for a fixed time slice in Anti de Sitter (AdS) space, mapping operators in the bulk of AdS, represented by logical qudits, onto operators on the boundary of the corresponding conformal field theory (CFT), represented by physical qudits. See [5; Defn. 4.3] for a technical formulation.


Protects against erasure errors on the boundary. Error-correction properties are often stated in the Heisenberg picture, i.e., in terms of which logical operators can be reconstructed after erasures. Specifically, bulk operators outside the entanglement wedges of the erased boundary operators can be reconstructed using the remaining boundary operators. However, the protection can be nontrivial, and may only apply to a subalgebra of bulk operators [6,7].

Typically, the encoding isometry \(U\) obeys the entanglement-wedge reconstruction condition, which states that for any boundary region \(R\), any bulk operator \(O\) localized to the entanglement wedge of \(R\) must be implementable by some boundary operator \(\tilde{O}\) localized to \(R\). Formally, \(UO = \tilde{O}U\) and \([\tilde{O},UU^\dagger] = 0\). The entanglement wedge is the space enclosed within the Ryu–Takayanagi surface in the bulk (minimal surface) with boundary \(R\).


Quantum encoding maps are isometries, but non-isometric encodings are relevant to describing mappings into the interior of a black hole [8] and de Sitter time evolution [9].

Transversal Gates

There exist holographic approximate codes with arbitrary transversal gate sets for any compact Lie group [5]. However, for sufficiently localized logical subsystems of holographic stabilizer codes, the set of transversally implementable logical operations is contained in the Clifford group [10].

Code Capacity Threshold

The ideal holographic tensor-network code (perfect representation of AdS/CFT) should be able to protect a central bulk operator against erasures of half of the physical qubits on the boundary, in line with AdS-Rindler reconstruction [1].Holographic tensor-network codes are argued to have a algebraic threshold, for which the error rate scales polynomially (as opposed to exponentially) in the thermodynamic limit [11]. Such a threshold is governed by the underlying conformal field theory describing the boundary.


  • Holographic code — Holographic codes whose encoders are holographic tensor networks are holographic tensor-network codes.
  • Quantum Lego code — Quantum Lego codes whose encoders are tensor networks discretizing hyperbolic space can be thought of as holographic codes. More generally, holograhpic tensor-network codes are types of quantum LEGO codes made from stabilizer codes where logical and physical legs are pre-assigned and logical legs are not contracted. In other words, logical legs resulting from the conversion of codes to tensors must remain logical in the final tensor network, and the same for physical. Contracting logical legs is another word for gluing two logical legs together.



  • Random stabilizer code — Random holographic tensor-network codes reproduce many aspects of holography [2,3,13].
  • Hamiltonian-based code — Local Hamiltonians lying at the CFT boundary can be mapped into the AdS bulk using tools from Hamiltonian simulation theory [14].
  • Hyperbolic surface code — Both holographic tensor-network and hyperbolic surface codes utilize tesselations of hyperbolic surfaces. Encodings for the former are hyperbolically tiled tensor networks, while the latter is defined on hyperbolically tiled physical-qubit lattices.


F. Pastawski et al., “Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence”, Journal of High Energy Physics 2015, (2015) arXiv:1503.06237 DOI
P. Hayden et al., “Holographic duality from random tensor networks”, Journal of High Energy Physics 2016, (2016) arXiv:1601.01694 DOI
X.-L. Qi and Z. Yang, “Space-time random tensor networks and holographic duality”, (2018) arXiv:1801.05289
T. Farrelly et al., “Tensor-Network Codes”, Physical Review Letters 127, (2021) arXiv:2009.10329 DOI
K. Dolev et al., “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
A. Almheiri, X. Dong, and D. Harlow, “Bulk locality and quantum error correction in AdS/CFT”, Journal of High Energy Physics 2015, (2015) arXiv:1411.7041 DOI
F. Pastawski and J. Preskill, “Code Properties from Holographic Geometries”, Physical Review X 7, (2017) arXiv:1612.00017 DOI
C. Akers et al., “The black hole interior from non-isometric codes and complexity”, (2022) arXiv:2207.06536
J. Cotler and A. Strominger, “The Universe as a Quantum Encoder”, (2022) arXiv:2201.11658
S. Cree et al., “Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted”, PRX Quantum 2, (2021) arXiv:2103.13404 DOI
N. Bao, C. Cao, and G. Zhu, “Deconfinement and error thresholds in holography”, Physical Review D 106, (2022) arXiv:2202.04710 DOI
D. Harlow, “The Ryu–Takayanagi Formula from Quantum Error Correction”, Communications in Mathematical Physics 354, 865 (2017) arXiv:1607.03901 DOI
H. Apel, T. Kohler, and T. Cubitt, “Holographic duality between local Hamiltonians from random tensor networks”, Journal of High Energy Physics 2022, (2022) arXiv:2105.12067 DOI
T. Kohler and T. Cubitt, “Toy models of holographic duality between local Hamiltonians”, Journal of High Energy Physics 2019, (2019) arXiv:1810.08992 DOI
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Zoo Code ID: holographic_tensor

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“Holographic tensor-network code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/holographic_tensor
@incollection{eczoo_holographic_tensor, title={Holographic tensor-network code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/holographic_tensor} }
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“Holographic tensor-network code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/holographic_tensor

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/block/tensor_network/holographic_tensor.yml.