## Description

A quantum code which encodes both quantum and classical information.

A simple example of a hybrid QECC encodes a single qubit and a single classical bit. A different quantum code \(\mathsf{C}_j\) is associated with each of the two values \(j\in\{0,1\}\) of the classical bit. The error words corresponding to correctable errors must satisfy the Knill-Laflamme conditions for each code [5; Eq. (3)], and error words beloning to different codes must be orthogonal to each other [5; Eq. (4)]. The corresponding decomposition of the Hilbert space \(\mathsf{H}\) is \begin{align} \mathsf{H} = \mathsf{C}_{1}\oplus\mathsf{C}_{2}\oplus\mathsf{C}^{\perp}~, \tag*{(1)}\end{align} where \(\mathsf{C}^\perp\) is the combined error space of both codes.

## Rate

## Parent

- Operator-algebra QECC (OAQECC) — An OAQECC which has no gauge structure (e.g., gauge qubits) but has a block structure that corresponds to a classical code is a hybrid QECC.

## Child

## Cousins

- Quantum error-correcting code (QECC) — A hybrid QECC storing no classical information reduces to a QECC. Conversely, any QECC can be converted into a hybrid QECC by using a portion of its logical subspace to store only classical information.
- Classical-quantum (c-q) code — A hybrid QECC storing no quantum information reduces to a c-q code.
- Entanglement-assisted (EA) hybrid quantum code — EQ hybrid codes utilize additional ancillary subsystems in a pre-shared entangled state, but reduce to hybrid QECCs when said subsystems are interpreted as noiseless physical subsystems.

## References

- [1]
- G. Kuperberg, “The capacity of hybrid quantum memory”, (2003) arXiv:quant-ph/0203105
- [2]
- I. Devetak and P. W. Shor, “The capacity of a quantum channel for simultaneous transmission of classical and quantum information”, (2004) arXiv:quant-ph/0311131
- [3]
- C. Bény, A. Kempf, and D. W. Kribs, “Generalization of Quantum Error Correction via the Heisenberg Picture”, Physical Review Letters 98, (2007) arXiv:quant-ph/0608071 DOI
- [4]
- 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
- [5]
- 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
- [6]
- D. W. Kribs et al., “Higher Rank Matricial Ranges and Hybrid Quantum Error Correction”, (2019) arXiv:1911.12744

## Page edit log

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

## Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/hybridqecc.yml.