## Description

A qubit stabilizer code which stores both quantum and classical information. Usually denoted as \([[n,k:c]]\) or \([[n,k:c,d]]\), where \(k\) (\(c\)) is the number of encoded qubits (classical bits), and where \(d\) is the distance.

The algebraic structure of a hybrid stabilizer code is the same as that of a USt code whose cosets are indexed by a linear binary code: both codes utilize codewords of an inner \([[n,k]]\) qubit stabilizer code \(\mathsf{C}\) and its cosets \(t \mathsf{C}\), where the \(2^c\) Pauli strings \(t\) correspond to the outer \([n,c]\) linear binary code. However, the hybrid stabilizer code does not utilize superpositions of codewords of \(t \mathsf{C}\) and \(t^{\prime} \mathsf{C}\) for \(t \neq t^{\prime}\) since the different coset blocks correspond to classical codewords.

## Parents

- Operator-algebra (OA) qubit stabilizer code — An OA stabilizer code which has no gauge qubits but has a block structure that corresponds to a linear binary code is a hybrid stabilizer code.
- Hybrid qubit code — An \([[n,k:c,d]]\) hybrid stabilizer code is an \(((n,2^k:2^c,d))\) hybrid qubit code.

## Children

## Cousins

- Qubit stabilizer code — A hybrid stabilizer code storing no classical information reduces to a qubit stabilizer code. Conversely, any qubit stabilizer code can be converted into a hybrid stabilizer code by using some its qubits to store only classical information [1].
- Union stabilizer (USt) code — The algebraic structure of a hybrid stabilizer code is the same as that of a USt code whose cosets are indexed by a linear binary code [1].
- \([[9,1,3]]\) Shor code — The Shor code can be modified to store three additional classical bits to yield a \([[9,1:3,3]]\) hybrid stabilizer code [1].
- \([[2m,2m-2,2]]\) error-detecting code — The \([[2m+1,2m+2:1,2]]\) hybrid stabilizer code [3] (extendable to modular qudits [4]) is closely related to the \([[2m,2m-2,2]]\) error-detecting code.
- \([[4,2,2]]\) Four-qubit code — The \([[4,2,2]]\) codewords can be modified by signs to yield a \([[4,1:1,2]]\) hybrid stabilizer code [5].
- Subsystem qubit stabilizer code — Hybrid stabilizer codes can be constructed from qubit subsystem stabilizer codes by using the gauge qubits of the latter to store classical information [6; Thm. 4].
- Subsystem hypergraph product (SHP) code — Hybrid stabilizer codes can be constructed from SHP codes by using the gauge qubits of the latter to store classical information [6; Sec. 4].
- EA qubit stabilizer code — EA hybrid stabilizer codes can be defined [1].
- Bacon-Shor code — There are several ways to convert Bacon-Shor codes to hybrid qubit stabilizer codes [6,7]

## References

- [1]
- I. Kremsky, M.-H. Hsieh, and T. A. Brun, “Classical enhancement of quantum-error-correcting codes”, Physical Review A 78, (2008) arXiv:0802.2414 DOI
- [2]
- M. Grassl, S. Lu, and B. Zeng, “Codes for simultaneous transmission of quantum and classical information”, 2017 IEEE International Symposium on Information Theory (ISIT) (2017) arXiv:1701.06963 DOI
- [3]
- A. Nemec and A. Klappenecker, “Infinite Families of Quantum-Classical Hybrid Codes”, (2020) arXiv:1911.12260
- [4]
- A. Nemec and A. Klappenecker, “Nonbinary Error-Detecting Hybrid Codes”, (2020) arXiv:2002.11075
- [5]
- S. Majidy, “A Unification of the Coding Theory and OAQEC Perspectives on Hybrid Codes”, International Journal of Theoretical Physics 62, (2023) arXiv:1806.03702 DOI
- [6]
- 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
- [7]
- 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 (2024-07-04) — most recent

## Cite as:

“Hybrid stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/hybrid_stabilizer