## 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 \([[n,k;e]]\) or \([[n,k,d;e]]\), where \(d\) is the distance of the underlying non-EA \([[n,k,d]]\) code, and \(e\) is the number of required pre-shared maximally entangled Bell states (ebits). 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.

The dual of an EA qubit stabilizer code is also an EA qubit stabilizer code whose logical qubits and ebits are interchanged, \(k\leftrightarrow e\) [3].

An \([[n,k+e;e]]\) 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 \(e = \text{rank}(AB^T+BA^T)\) [4].

## Rate

## Encoding

## Decoding

## Notes

## Parents

- EA qubit code
- EA Galois-qudit stabilizer code — EA Galois-qudit stabilizer codes reduce to EA qubit stabilizer codes for \(q=2\).

## Children

- \([[3, 1, 3;2]]\) EA code
- EA QLDPC code
- EA quantum convolutional code
- EA quantum turbo code
- 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 [12].

## Cousins

- 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. Qubit stabilizer codes can be used to obtain shortened EA qubit stabilizer codes [13].
- Hybrid stabilizer code — EA hybrid stabilizer codes can be defined [14].
- Linear binary code — Any linear binary code can be used to construct an EA qubit stabilizer code [1,2,5].
- Linear \(q\)-ary code — Any linear quaternary linear code can be used to construct an EA qubit stabilizer code [2].
- Qubit CSS code — As opposed to CSS codes, EA qubit stabilizer codes can be constructed from any linear binary code.
- Hybrid qudit-oscillator code — A minimal EA qubit stabilizer code has been realized in using hyper-entangled states [7].
- Asymmetric quantum code — Entanglement can help decode asymmetric quantum codes [15].
- \([[7,1,3]]\) Steane code — The Steane code is globally equivalent to a \([[6,1,3;1]]\) code, which is the smallest EA CSS code with that distance [16].
- Purity-testing stabilizer code — Purity-testing stabilizer codes are relevant to testing the purity of an entangled Bell state stabilized by two parties [17].
- Quantum Reed-Muller code — EA versions of quantum RM codes and their quantum tensor-product variants can be constructed [18].

## 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]
- C.-Y. Lai, T. A. Brun, and M. M. Wilde, “Dualities and identities for entanglement-assisted quantum codes”, Quantum Information Processing 13, 957 (2013) arXiv:1010.5506 DOI
- [4]
- M. M. Wilde and T. A. Brun, “Optimal entanglement formulas for entanglement-assisted quantum coding”, Physical Review A 77, (2008) arXiv:0804.1404 DOI
- [5]
- J. Qian and L. Zhang, “Entanglement-assisted quantum codes from arbitrary binary linear codes”, Designs, Codes and Cryptography 77, 193 (2014) DOI
- [6]
- A. K. Sharma and S. S. Garani, “Fault-Tolerant Quantum LDPC Encoders”, (2024) arXiv:2405.07242
- [7]
- M. M. Wilde and D. B. Uskov, “Linear-optical hyperentanglement-assisted quantum error-correcting code”, Physical Review A 79, (2009) arXiv:0807.4906 DOI
- [8]
- M. Grassl, “Entanglement-assisted quantum communication beating the quantum Singleton bound”, Physical Review A 103, (2021) arXiv:2007.01249 DOI
- [9]
- G. Luo et al., “Constructing quantum error-correcting codes that require a variable amount of entanglement”, Quantum Information Processing 23, (2023) arXiv:2207.05647 DOI
- [10]
- G. Luo et al., “Constructing quantum error-correcting codes that require a variable amount of entanglement”, Quantum Information Processing 23, (2023) DOI
- [11]
- C. H. Bennett et al., “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996) arXiv:quant-ph/9604024 DOI
- [12]
- 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
- [13]
- D. Ueno and R. Matsumoto, “Explicit method to make shortened stabilizer EAQECC from stabilizer QECC”, (2022) arXiv:2205.13732
- [14]
- 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
- [15]
- Y. Fujiwara and M.-H. Hsieh, “Adaptively correcting quantum errors with entanglement”, (2011) arXiv:1104.5004
- [16]
- B. Shaw et al., “Encoding one logical qubit into six physical qubits”, Physical Review A 78, (2008) arXiv:0803.1495 DOI
- [17]
- H. Barnum et al., “Authentication of quantum messages”, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings. arXiv:quant-ph/0205128 DOI
- [18]
- P. J. Nadkarni et al., “Entanglement-assisted Quantum Reed-Muller Tensor Product Codes”, Quantum 8, 1329 (2024) arXiv:2303.08294 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