Description
Family of hybrid qubit codes derived from Bacon-Shor subsystem codes by using their extra gauge structure to store classical information.
In the earlier gauge-fixing construction, one fixes commuting gauge operators and uses the corresponding gauge qubits to encode classical messages, yielding hybrid stabilizer codes with separate quantum and classical distances [1]. For the symmetric \(n\times n\) Bacon-Shor code, this gives the family \([[n^2,1:(n-1)^2,n:2]]_2\), including the \([[9,1:4,3:2]]_2\) code.
In the later operator-algebra construction, one instead chooses a nontrivial subset \(\mathcal{T}_0\) of coset representatives of the normalizer of the Bacon-Shor stabilizer group inside the Pauli group, so that the classical information is carried by distinct normalizer-coset sectors [2]. For the symmetric \(\ell\times\ell\) Bacon-Shor code, choosing \(\mathcal{T}_0\) generated by \(\prod_{i=1}^{\lfloor \ell / 2\rfloor} X_{(2i,1)}\) and \(\prod_{j=1}^{\lfloor \ell / 2\rfloor} Z_{(1,2j-1)}\) yields a \([[\ell^2,1:2,\lceil (\ell-1)/2 \rceil]]\) hybrid Bacon-Shor code, whose smallest nontrivial error-correcting member is the \([[16,1:2,2]]\) code.
Protection
In the gauge-fixing construction, the underlying Bacon-Shor subsystem code retains its quantum distance while the classical distance is determined by the chosen gauge fixing; in particular, the symmetric \(n\times n\) family above has parameters \([[n^2,1:(n-1)^2,n:2]]_2\) [1].
In the operator-algebra construction, for the canonical stabilizer generators of the symmetric \(\ell\times\ell\) Bacon-Shor code, each single-qubit Pauli error anti-commutes with at most two \(X\)-type and two \(Z\)-type stabilizer generators. Therefore, choosing \(X\)-type and \(Z\)-type coset representatives from classical linear codes \(C_X\) and \(C_Z\) with parameters \([\ell-1,k_X,d_X]\) and \([\ell-1,k_Z,d_Z]\), respectively, yields a hybrid Bacon-Shor code with \(|\mathcal{T}_0|=2^{k_X+k_Z}\) sectors and distance at least \begin{align} \min\left(\ell,\left\lceil d_X/2 \right\rceil,\left\lceil d_Z/2 \right\rceil\right)~. \tag*{(1)}\end{align} Choosing both \(C_X\) and \(C_Z\) to be the \([\ell-1,1,\ell-1]\) repetition code saturates this construction and yields the \([[\ell^2,1:2,\lceil (\ell-1)/2 \rceil]]\) family. For \(\ell=8\), taking both \(C_X\) and \(C_Z\) to be the \([7,4,3]\) Hamming code yields a \([[64,1:8,2]]\) hybrid Bacon-Shor code [2].
Cousin
- Bacon-Shor code— Hybrid Bacon-Shor codes are obtained from Bacon-Shor subsystem codes either by gauge fixing gauge qubits into classical registers [1] or by promoting a nontrivial subset of normalizer cosets to classical sectors [2].
Member of code lists
Primary Hierarchy
References
- [1]
- A. Nemec and A. Klappenecker, “Encoding classical information in gauge subsystems of quantum codes”, International Journal of Quantum Information 20, (2022) arXiv:2012.05896 DOI
- [2]
- G. Dauphinais, D. W. Kribs, and M. Vasmer, “Stabilizer Formalism for Operator Algebra Quantum Error Correction”, Quantum 8, 1261 (2024) arXiv:2304.11442 DOI
Page edit log
- Victor V. Albert (2026-03-27) — most recent
Cite as:
“Hybrid Bacon-Shor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/hybrid_bacon_shor