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].
Protection
Ancillary shared entanglement is assumed to be perfect, but this assumption can be relaxed [5]. There are quantum Griesmer [6] and Plotkin [7] bounds for EA qubit stabilizer codes.Rate
Asymptotically good EA qubit stabilizer codes exist [8].Encoding
Fault-tolerant encodes utilizing pre-shared entanglement [9].Decoding
Optical implementation of a minimal code using hyper-entangled states [10].Notes
Tables of bounds and examples of EA qubit (and EA qutrit) stabilizer codes for various \(n\) and \(k\), based on algorithms developed in Refs. [11–13], are maintained by M. Grassl at this website.See Ref. [12] for code tables and bounds on performance.See Ref. [14] for related notions.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 [15].
- Hybrid stabilizer code— EA hybrid stabilizer codes can be defined [16].
- Linear binary code— Any linear binary code can be used to construct an EA qubit stabilizer code [1,2,8].
- 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 [10].
- Concatenated qubit code— There exist concatenated EA qubit stabilizer codes that saturate the EA quantum Griesmer and Plotkin bounds [17].
- EA MDS code— There exist concatenated EA qubit stabilizer codes that saturate the EA quantum Singleton bound [17].
- Additive \(q\)-ary code— There is a relation between quaternary additive codes and EA qubit stabilizer codes [18].
- Asymmetric quantum code— Entanglement can help decode asymmetric quantum codes [19].
- \([[7,1,3]]\) 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 [20].
- Purity-testing stabilizer code— Purity-testing stabilizer codes are relevant to testing the purity of an entangled Bell state stabilized by two parties [21].
- Quantum Reed-Muller code— EA versions of quantum RM codes and their quantum tensor-product variants can be constructed [22].
Primary Hierarchy
Parents
EA Galois-qudit stabilizer codes reduce to EA qubit stabilizer codes for \(q=2\).
EA qubit stabilizer code
Children
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 [23].
References
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- 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]
- Y. Fujiwara, “Quantum Error Correction via Less Noisy Qubits”, Physical Review Letters 110, (2013) arXiv:1302.5081 DOI
- [6]
- R. Li, X. Li, and L. Guo, “On entanglement-assisted quantum codes achieving the entanglement-assisted Griesmer bound”, Quantum Information Processing 14, 4427 (2015) DOI
- [7]
- L. Guo and R. Li, “Linear Plotkin bound for entanglement-assisted quantum codes”, Physical Review A 87, (2013) DOI
- [8]
- J. Qian and L. Zhang, “Entanglement-assisted quantum codes from arbitrary binary linear codes”, Designs, Codes and Cryptography 77, 193 (2014) DOI
- [9]
- A. K. Sharma and S. S. Garani, “Fault-Tolerant Quantum LDPC Encoders”, (2024) arXiv:2405.07242
- [10]
- M. M. Wilde and D. B. Uskov, “Linear-optical hyperentanglement-assisted quantum error-correcting code”, Physical Review A 79, (2009) arXiv:0807.4906 DOI
- [11]
- M. Grassl, “Entanglement-assisted quantum communication beating the quantum Singleton bound”, Physical Review A 103, (2021) arXiv:2007.01249 DOI
- [12]
- G. Luo, M. F. Ezerman, M. Grassl, and S. Ling, “Constructing quantum error-correcting codes that require a variable amount of entanglement”, Quantum Information Processing 23, (2023) arXiv:2207.05647 DOI
- [13]
- G. Luo, M. F. Ezerman, M. Grassl, and S. Ling, “Constructing quantum error-correcting codes that require a variable amount of entanglement”, Quantum Information Processing 23, (2023) DOI
- [14]
- C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996) arXiv:quant-ph/9604024 DOI
- [15]
- D. Ueno and R. Matsumoto, “Explicit method to make shortened stabilizer EAQECC from stabilizer QECC”, (2022) arXiv:2205.13732
- [16]
- 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
- [17]
- N. R. Dash, S. Dutta, R. Srikanth, and S. Banerjee, “Bounds on concatenated entanglement-assisted quantum error-correcting codes”, (2024) arXiv:2412.16082
- [18]
- R. Li, Y. Ren, C. Guan, and Y. Liu, “Geometry of symplectic group and optimal EAQECC codes”, (2025) arXiv:2501.15465
- [19]
- Y. Fujiwara and M.-H. Hsieh, “Adaptively correcting quantum errors with entanglement”, (2011) arXiv:1104.5004
- [20]
- B. Shaw, M. M. Wilde, O. Oreshkov, I. Kremsky, and D. A. Lidar, “Encoding one logical qubit into six physical qubits”, Physical Review A 78, (2008) arXiv:0803.1495 DOI
- [21]
- H. Barnum, C. Crepeau, D. Gottesman, A. Smith, and A. Tapp, “Authentication of quantum messages”, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings. arXiv:quant-ph/0205128 DOI
- [22]
- P. J. Nadkarni, P. Jayakumar, A. Behera, and S. S. Garani, “Entanglement-assisted Quantum Reed-Muller Tensor Product Codes”, Quantum 8, 1329 (2024) arXiv:2303.08294 DOI
- [23]
- J. M. Renes, D. Sutter, F. Dupuis, and R. Renner, “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