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\([[7,2,2]]\) HGP phantom code[1]

Description

Smallest member of a family of CSS phantom HGP codes obtained from the hypergraph product of a classical simplex code and a repetition code.

A stabilizer tableau for an equivalent code is given by [2; ID 521] \begin{align} \begin{array}{ccccccc} Z & Z & I & I & I & I & Z \\ I & I & Z & I & Z & I & Z \\ I & I & I & Z & I & Z & Z \\ I & X & X & X & I & I & X \\ X & I & I & I & X & X & X \end{array}~. \tag*{(1)}\end{align}

Cousins

  • \([2^m-1,m,2^{m-1}]\) simplex code— This code is constructed from the hypergraph product of the \([3,2,2]\) simplex code and the \([2,1,2]\) repetition code [1].
  • Repetition code— This code is constructed from the hypergraph product of the \([3,2,2]\) simplex code and the \([2,1,2]\) repetition code [1].

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
Phantom code
Parents
Phantom code
The \([[7,2,2]]\) HGP code is the smallest member of a simplex/repetition HGP family of CSS phantom codes [1].
Hypergraph product (HGP) codeQLDPC CSS Generalized homological-product Lattice stabilizer Stabilizer Hamiltonian-based Qubit QECC Quantum
This code is the hypergraph product of the \([3,2,2]\) simplex code and the \([2,1,2]\) repetition code [1].
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[7,2,2]]\) HGP phantom code

References

[1]
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
[2]
Qiskit Community. Qiskit QEC framework. https://github.com/qiskit-community/qiskit-qec
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Zoo Code ID: hgp_7_2_2

Cite as:
“\([[7,2,2]]\) HGP phantom code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/hgp_7_2_2
BibTeX:
@incollection{eczoo_hgp_7_2_2, title={\([[7,2,2]]\) HGP phantom code}, booktitle={The Error Correction Zoo}, year={2026}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hgp_7_2_2} }
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Permanent link:
https://errorcorrectionzoo.org/c/hgp_7_2_2

Cite as:

“\([[7,2,2]]\) HGP phantom code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/hgp_7_2_2

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/7/hgp_7_2_2.yml.