\([[7,3,2]]\) punctured hypercube 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. It is the punctured version of the \([[8,3,2]]\) hypercube quantum code and is a phantom code [2].
A CSS stabilizer tableau for the code is given by [1; Table 8.6][3; ID 475] \begin{align} \begin{array}{ccccccc} Z & Z & Z & I & I & I & Z \\ I & Z & I & Z & I & Z & Z \\ Z & Z & I & I & Z & Z & I \\ X & X & X & X & X & X & X \end{array}~. \tag*{(1)}\end{align}
Protection
Distance two. As the \(k=3\) punctured hypercube code, it saturates the general qubit phantom-code bound \(n\geq 2^k-1\) [4].Cousins
- \([[8,3,2]]\) Smallest interesting color code— The \([[7,3,2]]\) code is obtained by puncturing one qubit from the \([[8,3,2]]\) hypercube quantum code [2].
- \([7,4,3]\) Hamming code— The \([[7,3,2]]\) punctured hypercube 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 \([[7,3,2]]\) punctured hypercube 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.
- \([[14,3,3]]\) CE phantom code— Concatenating the \([[7,3,(d_X=3,d_Z=2)]]\) punctured hypercube code with the two-qubit phase-flip repetition code yields this \([[14,3,(d_X=3,d_Z=4)]]\) CSS phantom code [2]. Dual-rail concatenation of the same punctured hypercube code yields a single-qubit Clifford-equivalent CE CSS frame [5].
Primary Hierarchy
Parents
This code is the punctured version of the \([[8,3,2]]\) hypercube quantum code and is a phantom code [2].
The punctured hypercube family is a quantum Reed-Muller family built from shortened and punctured classical Reed-Muller codes [4].
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[7,3,2]]\) punctured hypercube code
References
- [1]
- D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
- [2]
- J. M. Koh, A. Gong, A. C. Diaconu, D. B. Tan, A. A. Geim, M. J. Gullans, N. Y. Yao, M. D. Lukin, and S. Majidy, “Entangling logical qubits without physical operations”, (2026) arXiv:2601.20927
- [3]
- Qiskit Community, “Qiskit QEC framework”, URL
- [4]
- A. S. Morris and D. Malz, “Constraints on phantom codes from automorphism group bounds”, (2026) arXiv:2604.15111
- [5]
- C.-Y. Lai, P.-H. Liou, and Y. Ouyang, “Fault-Tolerant Quantum Error Correction for Constant-Excitation Stabilizer Codes under Coherent Noise”, (2025) arXiv:2507.10395
Page edit log
- Victor V. Albert (2026-05-12) — most recent
Cite as:
“\([[7,3,2]]\) punctured hypercube code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/xz_7_3_2