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\([[8,2,3]]\) Hermitian code[1]

Description

A non-CSS Hermitian eight-qubit stabilizer code that is the only Hermitian code and that has the largest automorphism group among the 20 inequivalent \([[8,2,3]]\) codes [2]. The code has an exceptionally short fault-tolerant syndrome measurement sequence [1,2].

A stabilizer tableau for the code is [3] \begin{align} \begin{array}{cccccccc} X & X & X & X & I & I & I & I \\ Z & Z & Z & Z & I & I & I & I \\ I & I & I & I & X & X & X & X \\ I & I & I & I & Z & Z & Z & Z \\ I & X & Y & Z & I & X & Y & Z \\ I & Z & X & Y & I & Z & X & Y \end{array}~, \tag*{(1)}\end{align} which is equivalent to [4; ID 4947].

Decoding

Short Shor-style syndrome extraction circuits can be used for syndrome-based decoding [1].

Fault Tolerance

Short Shor-style syndrome extraction circuits can be used for syndrome-based decoding [1].

Cousins

  • Concatenated qubit code— Applying the BLT mapping to the \([[8,2,3]]\) Hermitian code and concatenating each qubit pair with the \([[4,2,2]]\) code yields a \([[32,4,6]]\) self-dual CSS code [3; Corr. 2].
  • Self-dual CSS code— Applying the BLT mapping to the \([[8,2,3]]\) Hermitian code and concatenating each qubit pair with the \([[4,2,2]]\) code yields a \([[32,4,6]]\) self-dual CSS code [3; Corr. 2].
  • \([[4,2,2]]\) Four-qubit code— Applying the BLT mapping to the \([[8,2,3]]\) Hermitian code and concatenating each qubit pair with the \([[4,2,2]]\) code yields a \([[32,4,6]]\) self-dual CSS code [3; Corr. 2].

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
Hermitian qubit codeStabilizer Hamiltonian-based QECC Quantum
Parents
Hermitian qubit code
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[8,2,3]]\) Hermitian code

References

[1]
N. Delfosse and B. W. Reichardt, “Short Shor-style syndrome sequences”, (2020) arXiv:2008.05051
[2]
A. Cross and D. Vandeth, “Small Binary Stabilizer Subsystem Codes”, (2025) arXiv:2501.17447
[3]
B. W. Reichardt, D. Aasen, and R. Chao, “Fire and ice: Partially fault-tolerant quantum computing with selective state filtering”, (2026) arXiv:2605.15344
[4]
Qiskit Community, “Qiskit QEC framework”, URL
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Zoo Code ID: stab_8_2_3

Cite as:
“\([[8,2,3]]\) Hermitian code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_8_2_3
BibTeX:
@incollection{eczoo_stab_8_2_3, title={\([[8,2,3]]\) Hermitian code}, booktitle={The Error Correction Zoo}, year={2026}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stab_8_2_3} }
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Permanent link:
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Cite as:

“\([[8,2,3]]\) Hermitian code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_8_2_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/8/stab_8_2_3.yml.