Hybrid qubit code

Hybrid qubit code[1,2]

Description

A qubit code which stores both quantum and classical information. Usually denoted as \(((n,K:M))\) or \(((n,K:M,d))\), where \(K\) is the dimension of the underlying quantum code, \(M\) is the size of the classical code, and \(d\) is the distance.

Protection

Any qubit code can be converted into a hybrid qubit code by using some of its logical qubits to store only classical information [1]. An \(((n,K:M))\) hybrid qubit code can detect more errors than an \(((n,KM))\) qubit code [3]. A hybrid Hamming bound has been constructed [4].

Quantum weight enumerators, quantum MacWilliams identities, and linear programming bounds have been extended to hybrid qubit codes [2,3,5].

Cousins

Qubit code— A hybrid qubit code storing no classical information reduces to a qubit code. Conversely, any qubit code can be converted into a hybrid qubit code by using some of its logical qubits to store only classical information [1].
Qubit c-q code— A hybrid qubit code storing no quantum information reduces to a qubit c-q code.

Member of code lists

Primary Hierarchy

Parents

An OA qubit code that has no gauge structure (e.g., gauge qubits) but has a block structure that corresponds to a classical code is a hybrid qubit code.

Hybrid QECCQuantum

Hybrid qubit code

Children

Hybrid stabilizer code

An \([[n,k:c,d]]\) hybrid stabilizer code is an \(((n,2^k:2^c,d))\) hybrid qubit code.

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) 1718 (2017) arXiv:1701.06963 DOI
[3]: A. Nemec and A. Klappenecker, “Hybrid Codes”, 2018 IEEE International Symposium on Information Theory (ISIT) 796 (2018) arXiv:1901.02913 DOI
[4]: 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
[5]: A. Nemec and A. Klappenecker, “Infinite Families of Quantum-Classical Hybrid Codes”, (2020) arXiv:1911.12260

Page edit log

Victor V. Albert (2024-07-04) — most recent

Cite as:

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

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