Heptagon holographic code

Heptagon holographic code[1]

Alternative Names: Holographic Steane code.

Description

Holographic tensor-network code constructed out of a network of encoding isometries of the Steane code. Depending on how the isometry tensors are contracted, there is a zero-rate and a finite-rate code family.

Decoding

Optimal erasure decoder [1].

Code Capacity Threshold

\(~33\%\) under erasures using optimal erasure decoder for the finite-rate family, and \(50\%\) for the zero-rate family [1].Depolarizing noise: \(9.4\%\) using tensor-network decoder, and \(\approx 7\%\) using integer optimization decoder [2].\(18.985\%\) against depolarizing noise for zero-rate code under tensor-network decoder [3].

Cousin

Planar-perfect-tensor 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.

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum

Parents

Qubit CSS code

Holographic tensor-network codeHQECC QECC Quantum

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.

Heptagon holographic code

Children

Concatenated Steane code

A recursively concatenated Steane code is a heptagon holographic code on a tree tensor network.

References

[1]: R. J. Harris, N. A. McMahon, G. K. Brennen, and T. M. Stace, “Calderbank-Shor-Steane holographic quantum error-correcting codes”, Physical Review A 98, (2018) arXiv:1806.06472 DOI
[2]: T. Farrelly, R. J. Harris, N. A. McMahon, and T. M. Stace, “Parallel decoding of multiple logical qubits in tensor-network codes”, (2020) arXiv:2012.07317
[3]: J. Fan, M. Steinberg, A. Jahn, C. Cao, and S. Feld, “Biased-Noise Thresholds of Zero-Rate Holographic Codes with Tensor-Network Decoding”, (2025) arXiv:2408.06232

Page edit log

Victor V. Albert (2026-06-08) — most recent
Victor V. Albert (2024-07-01)

Cite as:

“Heptagon holographic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/holographic_steane, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/holographic/holographic_steane.yml.