Holographic code

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

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

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)

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.