# Holographic code[1]

## Description

A code whose encoding isometry serves to model aspects of the AdS/CFT holographic duality. Encodes operators in the bulk of the Anti de Sitter (AdS) space, represented by logical qudits, into operators on the boundary of the corresponding Conformal Field Theory (CFT), represented by physical qudits. Encoding can often be represented by a tensor network associated with a tiling of hyperbolic space. See Defn 4.3 of Ref. [2] for a technical formulation.

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 \(R\).

## Protection

## Transversal Gates

## Code Capacity Threshold

## Notes

## Parent

- Operator-algebra error-correcting code — Properties of holographic codes are often quantified in the Heisenberg picture, i.e., in terms of operator algebras [3][4].

## Children

- Pastawski-Yoshida-Harlow-Preskill (HaPPY) code
- Three qutrit code — Three-qutrit code is a minimal model for holography [3][8].

## Cousins

- Hyperbolic surface code — Both holographic 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.
- Quantum Lego code — Holographic codes whose encoders are tensor networks discretizing hyperbolic space are quantum Lego codes.

## 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). DOI; 1503.06237
- [2]
- K. Dolev et al., “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022). DOI; 2108.11402
- [3]
- A. Almheiri, X. Dong, and D. Harlow, “Bulk locality and quantum error correction in AdS/CFT”, Journal of High Energy Physics 2015, (2015). DOI; 1411.7041
- [4]
- F. Pastawski and J. Preskill, “Code Properties from Holographic Geometries”, Physical Review X 7, (2017). DOI; 1612.00017
- [5]
- S. Cree et al., “Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted”, PRX Quantum 2, (2021). DOI; 2103.13404
- [6]
- Ning Bao, ChunJun Cao, and Guanyu Zhu, “Deconfinement and Error Thresholds in Holography”. 2202.04710
- [7]
- Daniel Harlow and Hirosi Ooguri, “Symmetries in quantum field theory and quantum gravity”. 1810.05338
- [8]
- D. Harlow, “The Ryu–Takayanagi Formula from Quantum Error Correction”, Communications in Mathematical Physics 354, 865 (2017). DOI; 1607.03901

## Page edit log

- Victor V. Albert (2022-03-08) — most recent
- Joel Rajakumar (2021-12-20)

## Zoo code information

## Cite as:

“Holographic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/holographic

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/properties/holographic.yml.