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\([[15,4,3]]\) gauge-fixed quantum Hamming code[1,2]

Description

A pure CSS code obtained from the \([[15,7,3]]\) quantum Hamming code by gauging out three logical qubits, promoting to stabilizers the logical operator \(X^{\otimes 15}\) together with two weight-five \(X\)-type logical operators.

The \(Z\)-type stabilizers are generated by the \([15,4,8]\) simplex code, while the \(X\)-type stabilizers are generated by the \([15,7,5]\) BCH code of designed distance five, which contains both the simplex code and the all-ones vector. A stabilizer tableau for the code, in a qubit ordering for which both classical codes are cyclic, is [2][1; ID 6705228319cca60cf657a8ee] \begin{align} \begin{smallmatrix} Z & I & I & I & Z & Z & Z & Z & I & Z & I & Z & Z & I & I \\ I & Z & I & I & I & Z & Z & Z & Z & I & Z & I & Z & Z & I \\ I & I & Z & I & I & I & Z & Z & Z & Z & I & Z & I & Z & Z \\ I & I & I & Z & Z & Z & Z & I & Z & I & Z & Z & I & I & Z \\ X & I & I & I & I & I & I & X & X & X & I & X & I & I & I \\ I & X & I & I & I & I & I & I & X & X & X & I & X & I & I \\ I & I & X & I & I & I & I & I & I & X & X & X & I & X & I \\ I & I & I & X & I & I & I & I & I & I & X & X & X & I & X \\ I & I & I & I & X & I & I & X & X & X & I & I & X & X & I \\ I & I & I & I & I & X & I & I & X & X & X & I & I & X & X \\ I & I & I & I & I & I & X & X & X & I & X & I & I & I & X \end{smallmatrix}~. \tag*{(1)}\end{align}

The simultaneous permutation automorphism group of the two classical codes coincides with the automorphism group of the BCH code alone, which automatically preserves the simplex code. It is isomorphic to \(\Gamma L(2,4)\), of order 360, and acts transitively but imprimitively on the qubits, with five blocks of three qubits corresponding to the cosets of \(GF(4)^{\times}\) in \(GF(16)^{\times}\).

Protection

Since the minimum weight of an \(X\)-type logical operator is 3 while that of a \(Z\)-type logical operator is 4, the code detects \(X\)-type errors on up to 2 qubits and \(Z\)-type errors on up to 3 qubits.

Transversal and Permutation-Based Gates

Transversal \(\sqrt{X}\) preserves the code [2].All logical Clifford gates can be realized as two-fold transversal gates, i.e., by depth-one circuits of two-local code-preserving physical gates [2].Each of the four logical \(\sqrt{X}\) gates and each of the six Hadamard-conjugated logical controlled-\(Z\) gates can be realized individually by a depth-one circuit of two-local code-preserving physical gates, so that every \(X\)-type diagonal logical Clifford gate is addressable [2].

Cousins

  • \([[15, 7, 3]]\) quantum Hamming code— The \([[15,4,3]]\) code is obtained from the \([[15,7,3]]\) quantum Hamming code by gauging out three logical qubits, promoting to stabilizers the logical operator \(X^{\otimes 15}\) together with two weight-five \(X\)-type logical operators.
  • \([[15,6,3]]\) gauge-fixed quantum Hamming code— The \([[15,4,3]]\) code is obtained from the \([[15,6,3]]\) code by gauging out the two logical qubits whose logical \(X\) operators are supported on weight-five codewords of the \([15,7,5]\) BCH code. The two codes have the same \(Z\)-type stabilizer group. Both codes have exactly 35 weight-3 \(X\)-type logical operators, corresponding to the weight-3 codewords of the \([15,11,3]\) Hamming code, while every \(Z\)-type logical operator has even weight at least 4.
  • \([2^m-1,m,2^{m-1}]\) simplex code— The \(Z\)-type stabilizers of the \([[15,4,3]]\) code are generated by the \([15,4,8]\) simplex code.
  • Binary BCH code— The \(X\)-type stabilizers of the \([[15,4,3]]\) code are generated by the \([15,7,5]\) BCH 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
Qubit BCH codeStabilizer Hamiltonian-based Qubit QECC Quantum
Both classical codes underlying the \([[15,4,3]]\) code are BCH codes: the \(X\)-type stabilizers are generated by the \([15,7,5]\) BCH code of designed distance five, and the \(Z\)-type stabilizers by the \([15,4,8]\) simplex code, the BCH code of designed distance eight.
Cyclic quantum codeQECC Quantum
The \([[15,4,3]]\) code is cyclic in a qubit ordering in which both underlying classical codes are cyclic.
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[15,4,3]]\) gauge-fixed quantum Hamming code

References

[1]
S. Burton, “qecdb.org: Quantum Error Correction Database”, URL
[2]
V. V. Albert, “Beyond transversality: structure of Clifford circuits for CSS codes”, (2026) arXiv:2608.05688
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Zoo Code ID: stab_15_4_3

Cite as:
“\([[15,4,3]]\) gauge-fixed quantum Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_15_4_3, arXiv:2606.11484
BibTeX:
@incollection{eczoo_stab_15_4_3,
title={\([[15,4,3]]\) gauge-fixed quantum Hamming 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/stab_15_4_3}
}
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Permanent link:
https://errorcorrectionzoo.org/c/stab_15_4_3

Cite as:

“\([[15,4,3]]\) gauge-fixed quantum Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_15_4_3, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/15/stab_15_4_3.yml.