Description
A pure CSS code obtained from the \([[15,7,3]]\) quantum Hamming code by promoting the logical operator \(X^{\otimes 15}\) to a stabilizer, i.e., by gauging out the corresponding logical qubit [1; Sec. IV.A]. Equivalently, the code is obtained from the tesseract color code by removing one qubit.
The \(Z\)-type stabilizers are generated by the \([15,4,8]\) simplex code, while the \(X\)-type stabilizers are generated by the \([15,5,7]\) punctured first-order RM code, i.e., the span of 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][3; ID 6705228819cca60cf657a8f4] \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 & X & I & X & I & I & X & X & I & X & X \\ I & X & I & I & I & X & X & X & X & I & X & I & X & X & I \\ I & I & X & I & I & I & X & X & X & X & I & X & I & X & X \\ I & I & I & X & I & X & I & I & X & X & I & X & X & X & I \\ I & I & I & I & X & I & X & I & I & X & X & I & X & X & X \end{smallmatrix}~. \tag*{(1)}\end{align}
The simultaneous permutation automorphism group of the two classical codes is isomorphic to \(GL(4,2)\), of order 20160, and acts transitively on the qubits [1].
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
There are 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.Qubit permutations combined with transversal CNOT gates realize the full logical linear group \(GL(6,2)\) on one code block and the full \(GL(12,2)\) on pairs of code blocks [1; Sec. IV.A].Transversal \(\sqrt{X}\) preserves the code [2] and implements an order-two logical gate that is a product of Hadamard-conjugated controlled-\(Z\) gates on nine pairs of logical qubits, up to a logical Pauli correction.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].Cousins
- \([[15, 7, 3]]\) quantum Hamming code— The \([[15,6,3]]\) code is obtained from the \([[15,7,3]]\) quantum Hamming code by gauging out the logical qubit whose logical \(X\) operator is \(X^{\otimes 15}\) [1; Sec. IV.A].
- \([[16,6,4]]\) Tesseract color code— The \([[15,6,3]]\) code is obtained from the tesseract color code by removing one qubit, which punctures the underlying first-order RM code on the \(X\) side and shortens it to the simplex code on the \(Z\) side.
- \([2^m-1,m,2^{m-1}]\) simplex code— The \(Z\)-type stabilizers of the \([[15,6,3]]\) code are generated by the \([15,4,8]\) simplex code, and its \(X\)-type stabilizers by the simplex code together with the all-ones vector.
Primary Hierarchy
References
- [1]
- M. Grassl and M. Roetteler, “Leveraging automorphisms of quantum codes for fault-tolerant quantum computation”, 2013 IEEE International Symposium on Information Theory 534 (2013) arXiv:1302.1035 DOI
- [2]
- V. V. Albert, “Beyond transversality: structure of Clifford circuits for CSS codes”, (2026) arXiv:2608.05688
- [3]
- S. Burton, “qecdb.org: Quantum Error Correction Database”, URL
Page edit log
- Victor V. Albert (2026-08-08) — most recent
Cite as:
“\([[15,6,3]]\) gauge-fixed quantum Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_15_6_3, arXiv:2606.11484