\([[14,3,3]]\) CSS phantom code

\([[14,3,3]]\) CSS phantom code[1]

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.

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 [1].

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 [1].The Hadamard-dual code admits fold-diagonal logical \(S_iS_j\) and \(CZ_{ij}\) gates [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 [1].
Quantum repetition code— The inner code in the construction is the two-qubit phase-flip 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

This \([[14,3,3]]\) code is a CSS phantom code obtained from the punctured hypercube code and the two-qubit phase-flip repetition 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 [1].

Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum

\([[14,3,3]]\) CSS 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

Page edit log

Victor V. Albert (2026-05-20) — most recent

Cite as:

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

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