Here is a list of codes related to entanglement-assisted (EA) quantum codes.

Code | Description |
---|---|

EA FG-QLDPC code | One of several EA QLDPC code families constructed from finite-geometry LDPC (FG-LDPC) codes. There exists a family that requires an amount of entanglement that vanishes linearly with the length of each code. |

EA Galois-qudit code | Galois-qudit code designed to utilize pre-shared entanglement between sender and receiver. |

EA Galois-qudit stabilizer code | A Galois-qudit stabilizer code constructed using a variation of the stabilizer formalism designed to utilize pre-shared entanglement between sender and receiver. A code is typically denoted as \([[n,k;e]]_q\) or \([[n,k,d;e]]_q\), where \(d\) is the distance of the underlying non-EA \([[n,k,d]]_q\) code, and \(e\) is the number of required pre-shared maximally entangled Galois-qudit maximally entangled states. |

EA MDS code | EA Galois-qudit code whose parameters make the EAQECC Singleton bound (a.k.a. qubit-ebit Singleton bound) [1; Thm. 6] become an equality. |

EA QC-QLDPC code | One of several EA QLDPC code families constructed from QC-LDPC codes. |

EA QLDPC code | EA qubit stabilizer code for which the number of sites participating in each stabilizer generator and the number of stabilizer generators that each site participates in are both bounded by a constant \(w\) as \(n\to\infty\) |

EA analog stabilizer code | Constructed using a variation of the analog stabilizer formalism designed to utilize pre-shared entanglement between sender and receiver. |

EA bosonic code | Bosonic code designed to utilize pre-shared entanglement between sender and receiver. |

EA combinatorial-design QLDPC code | One of several EA QLDPC code families constructed from combinatorial designs. |

EA quantum LCD code | An EA Galois-qudit stabilizer code constructed from an LCD code. This family include the first asymptotically good EA Galois-qudit codes. |

EA quantum convolutional code | A quantum convolutional code designed to utilize pre-shared entanglement between sender and receiver, which can reduce memory requirements [2].' |

EA quantum turbo code | A quantum turbo code which uses pre-shared entanglement. This allows its encoder to be both recursive and non-catastrophic. |

EA qubit code | Qubit code designed to utilize pre-shared entanglement between sender and receiver. |

EA qubit stabilizer code | 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. |

Entanglement-assisted (EA) QECC | QECC whose encoding and decoding utilizes pre-shared entanglement between sender and receiver. |

Entanglement-assisted (EA) hybrid quantum code | Code that encodes quantum and classical information and requires pre-shared entanglement for transmission. |

Entanglement-assisted (EA) subsystem QECC | Subsystem QECC whose encoding and decoding utilizes pre-shared entanglement between sender and receiver. |

Error-corrected sensing code | Code that can be obtained via an optimization procedure that ensures correction against a set \(\cal{E}\) of errors as well as guaranteeting optimal precision in locally estimating a parameter using a noiseless ancilla. For tensor-product spaces consisting of \(n\) subsystems (e.g., qubits, modular qudits, or Galois qudits), the procedure can yield a code whose parameter estimation precision satisfies Heisenberg scaling, i.e., scales quadratically with the number \(n\) of subsystems. |

Maximal-entanglement EA Galois-qudit stabilizer code | An \([[n,k,d;e]]_q\) EA Galois-qudit stabilizer code for which \(e = n-k\). |

Quantum error-correcting code (QECC) | Encodes quantum information in a (logical) subspace of a (physical) Hilbert space such that it is possible to recover said information from errors that act as linear maps on the physical space. |

Quantum polar code | Entanglement-assisted CSS code utilized in a quantum polar coding scheme producing entangled pairs of qubits between sender and receiver. In such a scheme, the amplitude and phase information of a quantum state is handled in complementary fashion [3] using an encoding based on classical polar codes. Variants of the initial scheme have been developed for degradable channels [4] and extended to arbitrary channels [5]. |

\([[3, 1, 3;2]]\) EA code | Distance-three EA stabilizer code encoding one logical qubit and using two ebits. |

\([[4,2,2]]\) Four-qubit code | Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error. |

## References

- [1]
- M. Grassl, F. Huber, and A. Winter, “Entropic Proofs of Singleton Bounds for Quantum Error-Correcting Codes”, IEEE Transactions on Information Theory 68, 3942 (2022) arXiv:2010.07902 DOI
- [2]
- M. M. Wilde and T. A. Brun, “Extra shared entanglement reduces memory demand in quantum convolutional coding”, Physical Review A 79, (2009) arXiv:0812.4449 DOI
- [3]
- J. M. Renes and J.-C. Boileau, “Physical underpinnings of privacy”, Physical Review A 78, (2008) arXiv:0803.3096 DOI
- [4]
- M. M. Wilde and J. M. Renes, “Quantum polar codes for arbitrary channels”, 2012 IEEE International Symposium on Information Theory Proceedings (2012) arXiv:1201.2906 DOI
- [5]
- M. M. Wilde and S. Guha, “Polar Codes for Degradable Quantum Channels”, IEEE Transactions on Information Theory 59, 4718 (2013) arXiv:1109.5346 DOI