Here is a list of codes related to concatenated quantum codes.

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

Auxiliary qubit mapping (AQM) code | A concatenation of the JW transformation code with a qubit stabilizer code. |

Codeword stabilized (CWS) code | A code defined using a cluster state and a set of \(Z\)-type Pauli strings defined by a binary classical code. |

Concatenated Steane code | A member of the family of \([[7^m,1,3^m]]\) CSS codes, each of which is a recursive level-\(m\) concatenatenation of the Steane code. This family is one of the first to admit a concatenated threshold [1–5]. |

Concatenated bosonic code | A concatenated code whose outer code is a bosonic code. In other words, a bosonic code that can be thought of as a concatenation of a possibly non-bosonic inner code and another bosonic outer code. |

Concatenated c-q code | A c-q code constructed out of two classical or quantum codes for the purposes of transmission of classical information over quantum channels. |

Concatenated code | A code whose encoding mapping is a composition of two mappings: first the message set is mapped onto the code space of the outer code, then each coordinate of the outer code is mapped on the code space of the inner code. In the basic construction, the outer code's alphabet is the finite field \(GF(p^m)\) and the \(m\)-dimensional inner code is over over the field \(GF(p)\). The construction is not limited to linear codes. |

Concatenated quantum code | A combination of two quantum codes, an inner code \(C\) and an outer code \(C^\prime\), where the physical subspace used for the inner code consists of the logical subspace of the outer code. In other words, first one encodes in the inner code \(C^\prime\), and then one encodes each of the physical registers of \(C^\prime\) in an outer code \(C\). An inner \(C = ((n_1,K,d_1))_{q_1}\) and outer \(C^\prime = ((n_2,q_1,d_2))_{q_2}\) block quantum code yield an \(((n_1 n_2, K, d \geq d_1d_2))_{q_2}\) concatenated block quantum code [6]. |

Concatenated qubit code | A concatenated code whose outer code is a qubit code. In other words, a qubit code that can be thought of as a concatenation of an arbitrary inner code and another qubit outer code. An inner \(C = ((n_1,K,d_1))\) and outer \(C^\prime = ((n_2,2,d_2))\) qubit code yield an \(((n_1 n_2, K, d \geq d_1d_2))\) concatenated qubit code. |

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. |

Galois-qudit GRS code | True \(q\)-Galois-qudit stabilizer code constructed from GRS codes via either the Hermitian construction [7–9] or the Galois-qudit CSS construction [10,11]. |

Galois-qudit RS code | An \([[n,k,n-k+1]]_q\) (with \(q>n\)) Galois-qudit CSS code constructed using two RS codes over \(GF(q)\). |

Group-based QPC | An \([[m r,1,\min(m,r)]]_G\) generalization of the QPC. |

Group-based quantum repetition code | An \([[n,1]]_G\) generalization of the quantum repetition code. |

Hierarchical code | Member of a family of \([[n,k,d]]\) qubit stabilizer codes resulting from a concatenation of a constant-rate QLDPC code with a rotated surface code. Concatenation allows for syndrome extraction to be performed on a 2D geometry while maintining a threshold at the expense of a logarithmically vanishing rate. The growing syndrome extraction circuit depth allows known bounds in the literature to be weakened [12,13]. |

Jordan-Wigner transformation code | A mapping between qubit Pauli strings and Majorana operators that can be thought of as a trivial \([[n,n,1]]\) code. The mapping is best described as converting a chain of \(n\) qubits into a chain of \(2n\) Majorana modes (i.e., \(n\) fermionic modes). It maps Majorana operators into Pauli strings of weight \(O(n)\). |

Modular-qudit CWS code | A CWS code for modular qudits, defined using a modular-qudit cluster state and a set of modular-qudit \(Z\)-type Pauli strings defined by a \(q\)-ary classical code over \(\mathbb{Z}_q\). |

Quantum multi-dimensional parity-check (QMDPC) code | High-rate low-distance CSS code whose qubits lie on a \(D\)-dimensional rectangle, with \(X\)-type stabilizer generators defined on each \(D-1\)-dimensional rectangle. The \(Z\)-type stabilizer generators are defined via permutations in order to commute with the \(X\)-type generators. |

Quantum parity code (QPC) | A \([[m_1 m_2,1,\min(m_1,m_2)]]\) CSS code family obtained from concatenating an \(m_1\)-qubit phase-flip repetition code with an \(m_2\)-qubit bit-flip repetition code. |

Quantum repetition code | Encodes \(1\) qubit into \(n\) qubits according to \(|0\rangle\to|\phi_0\rangle^{\otimes n}\) and \(|1\rangle\to|\phi_1\rangle^{\otimes n}\). The code is called a bit-flip code when \(|\phi_i\rangle = |i\rangle\), and a phase-flip code when \(|\phi_0\rangle = |+\rangle\) and \(|\phi_1\rangle = |-\rangle\). |

Quantum turbo code | A quantum version of the turbo code, obtained from an interleaved serial quantum concatenation [14; Def. 30] of quantum convolutional codes. |

Rotated surface code | Variant of the surface code defined on a square lattice that has been rotated 45 degrees such that qubits are on vertices, and both \(X\)- and \(Z\)-type check operators occupy plaquettes in an alternating checkerboard pattern. |

Rotor GKP code | GKP code protecting against small angular position and momentum shifts of a planar rotor. |

Surface-17 code | A \([[9,1,3]]\) rotated surface code named for the sum of its 9 data qubits and 8 syndrome qubits. It uses the smallest number of qubits to perform fault-tolerant error correction on a surface code with parallel syndrome extraction. |

Yoked surface code | Member of a family of \([[n,k,d]]\) qubit CSS codes resulting from a concatenation of a QMDPC code with a rotated surface code. Concatenation does not impose additional connectivity constraints and can as much as triple the number of logical qubits per physical qubit when compared to the original surface code. Concatenation with 1D (2D) QMDPC yields codes with twice (four times) the distance. The stabilizer generators of the outer QMDPC code are referred to as yokes in this context. |

\([[2m,2m-2,2]]\) error-detecting code | Self-complementary CSS code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. The code is constructed via the CSS construction from an SPC code and a repetition code [15; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [16]. |

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

\([[4,2,2]]_{G}\) four group-qudit code | \([[4,2,2]]_{G}\) group quantum code that is an extension of the four-qubit code to group-valued qudits. |

\([[5,1,2]]\) rotated surface code | Rotated surface code on one rung of a ladder, with one qubit on the rung, and four qubits surrounding it. |

\([[6,4,2]]\) error-detecting code | Error-detecting six-qubit code with rate \(2/3\) whose codewords are cat/GHZ states. A set of stabilizer generators is \(XXXXXX\) and \(ZZZZZZ\). It is the unique code for its parameters, up to local equivalence [16; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [17]. |

\([[7,1,3]]\) Steane code | A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [18]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors. |

\([[9,1,3]]\) Shor code | Nine-qubit CSS code that is the first quantum error-correcting code. |

\([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code | An analog stabilizer version of Shor's nine-qubit code, encoding one mode into nine and correcting arbitrary errors on any one mode. |

\([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code | Modular-qudit CSS code that generalizes the \([[9,1,3]]\) Shor code using properties of the multiplicative group \(\mathbb{Z}_q\). |

## References

- [1]
- E. Knill, R. Laflamme, and W. H. Zurek, “Resilient quantum computation: error models and thresholds”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 365 (1998) arXiv:quant-ph/9702058 DOI
- [2]
- A. M. Steane, “Efficient fault-tolerant quantum computing”, Nature 399, 124 (1999) arXiv:quant-ph/9809054 DOI
- [3]
- A. M. Steane, “Overhead and noise threshold of fault-tolerant quantum error correction”, Physical Review A 68, (2003) arXiv:quant-ph/0207119 DOI
- [4]
- K. M. Svore, B. M. Terhal, and D. P. DiVincenzo, “Local fault-tolerant quantum computation”, Physical Review A 72, (2005) arXiv:quant-ph/0410047 DOI
- [5]
- K. M. Svore, D. P. DiVincenzo, and B. M. Terhal, “Noise Threshold for a Fault-Tolerant Two-Dimensional Lattice Architecture”, (2006) arXiv:quant-ph/0604090
- [6]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [7]
- L. Jin and C. Xing, “A Construction of New Quantum MDS Codes”, (2020) arXiv:1311.3009
- [8]
- X. Liu, L. Yu, and H. Liu, “New quantum codes from Hermitian dual-containing codes”, International Journal of Quantum Information 17, 1950006 (2019) DOI
- [9]
- L. Jin et al., “Application of Classical Hermitian Self-Orthogonal MDS Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 56, 4735 (2010) DOI
- [10]
- D. Aharonov and M. Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”, (1999) arXiv:quant-ph/9906129
- [11]
- Z. Li, L.-J. Xing, and X.-M. Wang, “Quantum generalized Reed-Solomon codes: Unified framework for quantum maximum-distance-separable codes”, Physical Review A 77, (2008) arXiv:0812.4514 DOI
- [12]
- N. Delfosse, M. E. Beverland, and M. A. Tremblay, “Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes”, (2021) arXiv:2109.14599
- [13]
- N. Baspin, O. Fawzi, and A. Shayeghi, “A lower bound on the overhead of quantum error correction in low dimensions”, (2023) arXiv:2302.04317
- [14]
- D. Poulin, J.-P. Tillich, and H. Ollivier, “Quantum serial turbo-codes”, (2009) arXiv:0712.2888
- [15]
- N. Rengaswamy et al., “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) (2018) arXiv:1803.06987 DOI
- [16]
- A. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
- [17]
- H. Goto, “Many-hypercube codes: High-rate quantum error-correcting codes for high-performance fault-tolerant quantum computing”, (2024) arXiv:2403.16054
- [18]
- B. Shaw et al., “Encoding one logical qubit into six physical qubits”, Physical Review A 78, (2008) arXiv:0803.1495 DOI