## Description

Constructed using a variation of the stabilizer formalism designed to utilize pre-shared entanglement between sender and receiver. A code is typically denoted as \([[n,k;c]]\) or \([[n,k,d;c]]\), where \(d\) is the distance of the underlying non-EA \([[n,k,d]]\) code, and \(c\) is the number of required pre-shared maximally entangled Bell states. While other entangled states can be used, there is always a choice a generators such that the Bell state suffices while still using the fewest ebits.

An \([[n,k+c;c]]\) EA stabilizer code can be constructed from an ordinary \([[n,k]]\) stabilizer code with check matrix \(H=(A|B)\), where the required number of ebits is \(c = \text{rank}(AB^T+BA^T)\) [3].

## Decoding

## Notes

## Parents

- Qubit stabilizer code — EA qubit stabilizer codes utilize additional ancillary qubits in a pre-shared entangled state, but reduce to qubit stabilizer codes when said qubits are interpreted as noiseless physical qubits.
- Entanglement-assisted (EA) QECC

## Child

- Quantum polar code — Quantum polar codes are CSS codes used in an entanglement generation scheme that generally requires entanglement assistance. They require assistance only to determine positions to store information which optimally protect against both bit and phase noise. Without this assistance, they are just CSS codes constructed out of polar codes. A variant of quantum polar codes exists that does not require entanglement assistance [7].

## Cousins

- Linear binary code — Any linear binary code can be used to construct an EA qubit stabilizer code.
- Linear \(q\)-ary code — Any linear quaternary (\(q=4\)) code can be used to construct an EA qubit stabilizer code.
- Calderbank-Shor-Steane (CSS) stabilizer code — As opposed to CSS codes, EA qubit stabilizer codes can be constructed from any linear binary code.

## References

- [1]
- T. A. Brun, I. Devetak, and M.-H. Hsieh, “Catalytic Quantum Error Correction”, IEEE Transactions on Information Theory 60, 3073 (2014) arXiv:quant-ph/0608027 DOI
- [2]
- T. Brun, I. Devetak, and M.-H. Hsieh, “Correcting Quantum Errors with Entanglement”, Science 314, 436 (2006) arXiv:quant-ph/0610092 DOI
- [3]
- M. M. Wilde and T. A. Brun, “Optimal entanglement formulas for entanglement-assisted quantum coding”, Physical Review A 77, (2008) arXiv:0804.1404 DOI
- [4]
- M. M. Wilde and D. B. Uskov, “Linear-optical hyperentanglement-assisted quantum error-correcting code”, Physical Review A 79, (2009) arXiv:0807.4906 DOI
- [5]
- G. Luo et al., “How Much Entanglement Does a Quantum Code Need?”, (2022) arXiv:2207.05647
- [6]
- C. H. Bennett et al., “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996) arXiv:quant-ph/9604024 DOI
- [7]
- J. M. Renes et al., “Efficient Quantum Polar Codes Requiring No Preshared Entanglement”, IEEE Transactions on Information Theory 61, 6395 (2015) arXiv:1307.1136 DOI

## Page edit log

- Victor V. Albert (2023-01-12) — most recent
- Lane G. Gunderman (2023-01-12)
- Victor V. Albert (2022-07-14)

## Cite as:

“EA qubit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/eastab

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/eastab.yml.