Holographic code[1] 

Description

Block quantum code whose features (typically, the encoding isometry) serve to model aspects of the AdS/CFT holographic duality.

The encoding map often 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. 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

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 [3,4].

Transversal Gates

There exist holographic approximate codes with arbitrary transversal gate sets for any compact Lie group [2]. However, for sufficiently localized logical subsystems of holographic stabilizer codes, the set of transversally implementable logical operations is contained in the Clifford group [5].

Code Capacity Threshold

The ideal holographic 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 codes are argued to have a algebraic threshold, for which the error rate scales polynomially (as opposed to exponentially) in the thermodynamic limit [6]. Such a threshold is governed by the underlying conformal field theory describing the boundary.

Notes

All Boundary global symmetries must be dual to bulk gauge symmetries, and vice versa [7].

Parent

Children

Cousins

  • Quantum Lego code — Quantum Lego codes codes whose encoders are tensor networks discretizing hyperbolic space can be thought of as holographic codes. More generally, tensor-network codes are types of 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.
  • Approximate quantum error-correcting code (AQECC) — Universal subspace approximate error correction is used to model black holes [12].
  • Approximate 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].
  • Concatenated quantum code — A holographic code whose encoding circuit is arranged in a tree geometry reduces to a concatenated code.
  • 2D lattice stabilizer code — 2D lattice stabilizer codes admit a bulk-boundary correspondence similar to that of holographic codes, namely, the boundary Hilbert space of the former cannot be realized via local degrees of freedom [13].
  • 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.

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]
K. Dolev et al., “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
[3]
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
[4]
F. Pastawski and J. Preskill, “Code Properties from Holographic Geometries”, Physical Review X 7, (2017) arXiv:1612.00017 DOI
[5]
S. Cree et al., “Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted”, PRX Quantum 2, (2021) arXiv:2103.13404 DOI
[6]
N. Bao, C. Cao, and G. Zhu, “Deconfinement and error thresholds in holography”, Physical Review D 106, (2022) arXiv:2202.04710 DOI
[7]
D. Harlow and H. Ooguri, “Symmetries in quantum field theory and quantum gravity”, (2019) arXiv:1810.05338
[8]
K. Furuya, N. Lashkari, and S. Ouseph, “Real-space RG, error correction and Petz map”, Journal of High Energy Physics 2022, (2022) arXiv:2012.14001 DOI
[9]
C. Cao, G. Cheng, and B. Swingle, “Large \(N\) Matrix Quantum Mechanics as a Quantum Memory”, (2022) arXiv:2211.08448
[10]
G. Bentsen, P. Nguyen, and B. Swingle, “Approximate Quantum Codes From Long Wormholes”, (2023) arXiv:2310.07770
[11]
D. Harlow, “The Ryu–Takayanagi Formula from Quantum Error Correction”, Communications in Mathematical Physics 354, 865 (2017) arXiv:1607.03901 DOI
[12]
P. Hayden and G. Penington, “Learning the Alpha-bits of black holes”, Journal of High Energy Physics 2019, (2019) arXiv:1807.06041 DOI
[13]
T. Schuster et al., “A holographic view of topological stabilizer codes”, (2023) arXiv:2312.04617
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Zoo Code ID: holographic

Cite as:
“Holographic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/holographic
BibTeX:
@incollection{eczoo_holographic, title={Holographic code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/holographic} }
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“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/edit/main/codes/quantum/properties/block/tensor_network/holographic.yml.