## Description

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.

## Protection

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 \(O^{\prime}\) localized to \(R\). Formally, \(UO = O^{\prime}U\) and \([O^{\prime},UU^\dagger] = 0\). The entanglement wedge is the space enclosed within the Ryu–Takayanagi surface in the bulk (minimal surface) with boundary \(R\).

## Encoding

## Transversal Gates

## Code Capacity Threshold

## Notes

## Parents

- Holographic code — Holographic codes whose encoders are holographic tensor networks are holographic tensor-network codes.
- Tensor-network 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.

## Children

- Pastawski-Yoshida-Harlow-Preskill (HaPPY) code — The encoding of a HaPPy code is a holographic tensor network consisting of pentagon and hexagon perfect tensors.
- Surface-code-fragment (SCF) holographic code — The encoding of the heptagon holographic code is a holographic tensor network consisting of the encoding isometry for the \([[5,1,2]]\) rotated surface code, which is a planar-perfect tensor.
- Six-qubit-tensor holographic code — The encoding of the six-qubit-tensor holographic code is a holographic tensor network consisting of the encoding isometry for the \([[6,1,3]]\) six-qubit stabilizer code.
- Hyperinvariant tensor-network (HTN) code — The encoding of an HTN code is a hyperinvariant tensor network.
- Heptagon holographic code — The encoding of the heptagon holographic code is a holographic tensor network consisting of the encoding isometry for the Steane code, which is a planar-perfect tensor.
- Holographic hybrid code — The holographic hybrid code is constructed out of alternating isometries of the five-qubit and \([[4,1,1,2]]\) Bacon-Shor codes.
- Three-qutrit code — The three-qutrit code is a radius-one holographic tensor-network code and serves as a minimal model for holography [6,14].

## Cousins

- Random stabilizer code — Random holographic tensor-network codes reproduce many aspects of holography [2,3,15].
- Hamiltonian-based code — Local Hamiltonians lying at the CFT boundary can be mapped into the AdS bulk using tools from Hamiltonian simulation theory [16].
- Galois-qudit GRS code — Galois-qudit GRS codes can be used to construct holographic p-adic (i.e., tree-tensor-network) codes on Bruhat-Tits trees and buildings and on Drinfeld symmetric spaces [17,18].
- 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.

## References

- [1]
- 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
- [2]
- P. Hayden et al., “Holographic duality from random tensor networks”, Journal of High Energy Physics 2016, (2016) arXiv:1601.01694 DOI
- [3]
- X.-L. Qi and Z. Yang, “Space-time random tensor networks and holographic duality”, (2018) arXiv:1801.05289
- [4]
- T. Farrelly et al., “Tensor-Network Codes”, Physical Review Letters 127, (2021) arXiv:2009.10329 DOI
- [5]
- K. Dolev et al., “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
- [6]
- 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
- [7]
- F. Pastawski and J. Preskill, “Code Properties from Holographic Geometries”, Physical Review X 7, (2017) arXiv:1612.00017 DOI
- [8]
- C. Akers et al., “The black hole interior from non-isometric codes and complexity”, (2022) arXiv:2207.06536
- [9]
- J. Cotler and A. Strominger, “The Universe as a Quantum Encoder”, (2022) arXiv:2201.11658
- [10]
- S. Cree et al., “Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted”, PRX Quantum 2, (2021) arXiv:2103.13404 DOI
- [11]
- N. Bao, C. Cao, and G. Zhu, “Deconfinement and error thresholds in holography”, Physical Review D 106, (2022) arXiv:2202.04710 DOI
- [12]
- A. May, G. Penington, and J. Sorce, “Holographic scattering requires a connected entanglement wedge”, Journal of High Energy Physics 2020, (2020) arXiv:1912.05649 DOI
- [13]
- H. Apel et al., “Security of quantum position-verification limits Hamiltonian simulation via holography”, Journal of High Energy Physics 2024, (2024) arXiv:2401.09058 DOI
- [14]
- D. Harlow, “The Ryu–Takayanagi Formula from Quantum Error Correction”, Communications in Mathematical Physics 354, 865 (2017) arXiv:1607.03901 DOI
- [15]
- 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
- [16]
- T. Kohler and T. Cubitt, “Toy models of holographic duality between local Hamiltonians”, Journal of High Energy Physics 2019, (2019) arXiv:1810.08992 DOI
- [17]
- M. Marcolli, “Holographic Codes on Bruhat--Tits buildings and Drinfeld Symmetric Spaces”, (2018) arXiv:1801.09623
- [18]
- M. Heydeman et al., “Nonarchimedean Holographic Entropy from Networks of Perfect Tensors”, (2018) arXiv:1812.04057

## Page edit log

- Matthew Steinberg (2024-07-01) — most recent
- Victor V. Albert (2024-07-01)

## Cite as:

“Holographic tensor-network code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/holographic_tensor