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\([[14,3,3]]\) CE phantom code[1,2]

Description

CSS phantom code obtained by concatenating the \([[7,3,(d_X=3,d_Z=2)]]\) punctured hypercube code with the two-qubit phase-flip repetition code. The code is equivalent to the \([[14,3,3]]\) constant-excitation (CE) CSS code obtained by applying dual-rail concatenation to the \([[7,3,2]]\) punctured hypercube code, up to single-qubit Clifford gates, a physical-qubit permutation, and a Pauli frame [1].

One stabilizer tableau for the code is \begin{align} \begin{smallmatrix} Z & Z & Z & Z & Z & Z & I & I & I & I & I & I & Z & Z \\ I & I & Z & Z & I & I & Z & Z & I & I & Z & Z & Z & Z \\ Z & Z & Z & Z & I & I & I & I & Z & Z & Z & Z & I & I \\ X & X & I & I & I & I & I & I & I & I & I & I & I & I \\ I & I & X & X & I & I & I & I & I & I & I & I & I & I \\ I & I & I & I & X & X & I & I & I & I & I & I & I & I \\ I & I & I & I & I & I & X & X & I & I & I & I & I & I \\ I & I & I & I & I & I & I & I & X & X & I & I & I & I \\ I & I & I & I & I & I & I & I & I & I & X & X & I & I \\ I & I & I & I & I & I & I & I & I & I & I & I & X & X \\ X & I & X & I & X & I & X & I & X & I & X & I & X & I \end{smallmatrix}~. \tag*{(1)}\end{align}

Protection

Corrects a single-qubit error. Its \(X\)- and \(Z\)-sector distances are \(d_X=3\) and \(d_Z=4\), respectively [2].

Gates

The code is phantom, so every ordered-pair in-block logical CNOT gate between its three logical qubits can be implemented by a physical-qubit permutation [2].The Hadamard-dual code admits fold-diagonal logical \(S_iS_j\) and \(CZ_{ij}\) gates [2].

Fault Tolerance

In the locally Clifford-equivalent CE CSS frame, fault-tolerant syndrome extraction can use modified Shor and Steane methods adapted for CE codes: weight-\(2w\) stabilizers are measured using \(w\)-CE cat states, and zero-controlled NOT (\(\mathrm{C}_0 X\)) gates replace standard CNOT gates to preserve the constant-excitation structure [1].

Cousins

  • \([[7,3,2]]\) punctured hypercube 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 [1].
  • \([[7,1,3]]\) Steane code— Dual-rail concatenation of the \([[7,1,3]]\) Steane code yields a \([[14,1,3]]\) CE CSS code, from which the locally Clifford-equivalent \([[14,3,3]]\) CE CSS frame is obtained by removing two independent \(Z\)-type stabilizer generators [1].
  • Quantum repetition code— The inner code in the construction is the two-qubit phase-flip repetition code [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
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum
Phantom code
Parents
Phantom code
This \([[14,3,3]]\) code is a CSS phantom code obtained from the punctured hypercube code and the two-qubit phase-flip repetition code [2].
Constant-excitation (CE) codeHamiltonian-based QECC Quantum
This code is single-qubit Clifford equivalent to the \([[14,3,3]]\) CE CSS code obtained by dual-rail concatenation of the \([[7,3,2]]\) punctured hypercube code [1].
Concatenated qubit codeQubit Concatenated quantum QECC Quantum
This code is a concatenation of the \([[7,3,2]]\) punctured hypercube code with the two-qubit phase-flip repetition code [2].
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[14,3,3]]\) CE phantom code

References

[1]
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
[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
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)

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Zoo Code ID: phantom_14_3_3

Cite as:
“\([[14,3,3]]\) CE phantom code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/phantom_14_3_3, arXiv:2606.11484
BibTeX:
@incollection{eczoo_phantom_14_3_3,
title={\([[14,3,3]]\) CE phantom code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/phantom_14_3_3}
}
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Permanent link:
https://errorcorrectionzoo.org/c/phantom_14_3_3

Cite as:

“\([[14,3,3]]\) CE phantom code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/phantom_14_3_3, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/14/phantom_14_3_3.yml.