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

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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 QECC 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. The state space of a QECC is contained in the space of complex \(L^2\)-normalizable functions of some configuration space, which usually corresponds to the alphabet of a classical code.
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
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Error correction zoo by Victor V. Albert, Philippe Faist, and many contributors. This work is licensed under a CC-BY-SA License. See how to contribute.