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\([[7,3,2]]\) \(X/Z\)-error-correcting code[1]

Description

A seven-qubit pure CSS code that corrects a single \(X\) or a single \(Z\) error, but not a single \(Y\) error.

A stabilizer tableau for the code is given by [1; Table 8.6][2; ID 475] \begin{align} \begin{array}{ccccccc} Z & Z & Z & Z & Z & Z & Z \\ Y & Y & Y & Y & I & I & I \\ Y & Y & I & I & Y & Y & I \\ Y & I & Y & I & Y & I & Y \end{array}~. \tag*{(1)}\end{align} The code is CSS, but the above non-CSS form is required to correct a single \(X\) or a single \(Z\) error.

Cousins

  • \([7,4,3]\) Hamming code— The \([[7,3,2]]\) \(X/Z\)-error-correcting code \(H_X\) check matrix is the parity-check matrix of the \([7,4,3]\) Hamming code, while its \(H_Z\) matrix is that of the SPC code.
  • \([n,n-1,2]\) Single parity-check (SPC) code— The \([[7,3,2]]\) \(X/Z\)-error-correcting code \(H_X\) check matrix is the parity-check matrix of the \([7,4,3]\) Hamming code, while its \(H_Z\) matrix is that of the SPC code.

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
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[7,3,2]]\) \(X/Z\)-error-correcting code

References

[1]
D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
[2]
Qiskit Community. Qiskit QEC framework. https://github.com/qiskit-community/qiskit-qec
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Zoo Code ID: xz_7_3_2

Cite as:
“\([[7,3,2]]\) \(X/Z\)-error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/xz_7_3_2
BibTeX:
@incollection{eczoo_xz_7_3_2, title={\([[7,3,2]]\) \(X/Z\)-error-correcting code}, booktitle={The Error Correction Zoo}, year={2026}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/xz_7_3_2} }
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Permanent link:
https://errorcorrectionzoo.org/c/xz_7_3_2

Cite as:

“\([[7,3,2]]\) \(X/Z\)-error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/xz_7_3_2

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/7/xz_7_3_2.yml.