Hybrid stabilizer code[1,2] 

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
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Zoo Code ID: hybrid_stabilizer

Cite as:
“Hybrid stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/hybrid_stabilizer
BibTeX:
@incollection{eczoo_hybrid_stabilizer, title={Hybrid stabilizer code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hybrid_stabilizer} }
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“Hybrid stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/hybrid_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/oa/hybrid_stabilizer.yml.