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

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

3D color code | Color code defined on a four-valent four-colorable tiling of 3D space. Logical dimension is determined by the genus of the underlying surface (for closed surfaces) and types of boundaries (for open surfaces). |

Auxiliary qubit mapping (AQM) code | A concatenation of the JW transformation code with a qubit stabilizer 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 inner code is a bosonic 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 combination of two codes, an inner code \(C\) and an outer code \(C^\prime\), where the coordinates used for the inner code consist of the message subspace of the outer code. |

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 \(C^\prime = ((n_2,q_1,d_2))\) block quantum code yield an \(((n_1 n_2, K, d \geq d_1d_2))_{q_2}\) concatenated quantum code [6]. |

Dual-rail quantum code | Two-mode bosonic code encoding a logical qubit in Fock states with one excitation. The logical-zero state is represented by \(|10\rangle\), while the logical-one state is represented by \(|01\rangle\). This encoding is often realized in temporal or spatial modes, corresponding to a time-bin or frequency-bin encoding. Two different types of photon polarization can also be used. |

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. |

Five-qubit perfect code | Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error. |

GKP cluster-state code | Multi-mode code encoding logical qubits into a cluster-state stabilizer code concatenated with a single-mode GKP code. Provides a way to perform a continuous-variable (CV) analogue of fault-tolerant MBQC. |

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

Gauss' law code | An \([m+Dm,Dm,3]\) linear binary code for \(m\geq 3^D\), defined by the Gauss' law constraint of a \(D\)-dimensional fermionic \(\mathbb{Z}_2\) gauge theory [12; Thm. 1]. The code can be re-phrased as a distance-one stabilizer code whose stabilizers consist of gauge-group elements. It can be concatenated to form a stabilizer code for fault-tolerant quantum simulation of the underlying gauge theory [12,13]. |

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 [14,15]. |

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 \(O(n)\). |

Layer code | Member of a family of 3D lattice CSS codes with stabilizer generator weights \(\leq 6\) that are obtained by coupling layers of 2D surface code according to the Tanner graph of a QLDPC code. Geometric locality is maintained because, instead of being concatenated, each pair of parallel surface-code squares are fused (or quasi-concatenated) with perpendicular surface-code squares via lattice surgery. |

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 divisible code | A level-\(\nu\) quantum divisible code is a CSS code whose \(X\)-type stabilizers form a \(\nu\)-even linear binary code in the symplectic representation and which admits a transversal gate at the \(\nu\)th level of the Clifford hierarchy. A CSS code is doubly-even (triply-even) if all \(X\)-type stabilizer generators have weight divisible by four (eight), i.e., if they form a doubly-even (triply-even) linear binary code. Doubly-even codes can yield a transversal \(S\) gate, while triply-even codes yield a transversal \(T\) gate for odd \(n\) [16]. |

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 \(|1\rangle\to|\phi_1\rangle^{\otimes n}\). Also known as 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 [17; Def. 30] of quantum convolutional codes. |

Raussendorf-Bravyi-Harrington (RBH) cluster-state code | A three-dimensional cluster-state code defined on the bcc lattice (i.e., a cubic lattice with qubits on edges and faces). |

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. |

Subsystem homological product code | A CSS subsystem code constructed from a product of two (subspace) CSS codes. The case for qubits is formulated below, but these codes have also been extended to Galois qudits [18]. |

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. |

Two-mode binomial code | Two-mode constant-energy CLY code whose coefficients are square-roots of binomial coefficients. |

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. |

\(((n,1,2))\) Bravyi-Lee-Li-Yoshida PI code | PI distance-two code on \(n\geq4\) qubits whose degree of entanglement vanishes asymptotically with \(n\) [19; Appx. D]. |

\([[15,1,3]]\) quantum Reed-Muller code | \([[15,1,3]]\) CSS code that is most easily thought of as a tetrahedral 3D color code. This code contains 15 qubits, represented by four vertices, four face centers, six edge centers, and one body center. The tetrahedron is cellulated into four identical polyhedron cells by connecting the body center to all four face centers, where each face center is then connected by three adjacent edge centers. Each colored cell corresponds to a weight-eight \(X\)-check, and each face corresponds to a weight-4 \(Z\)-check. A logical \(Z\) is any weight-3 \(Z\)-string along an edge of the entire tetrahedron. The logical \(X\) is any weight-7 \(X\)-face of the entire tetrahedron. |

\([[2^D,D,2]]\) hypercube code | Member of a family of codes defined by placing qubits on a \(D\)-dimensional hypercube, \(Z\)-stabilizers on all two-dimensional faces, and an \(X\)-stabilizer on all vertices. These codes realize gates at the \((D-1)\)-st level of the Clifford hierarchy. It can be generalized to a \([[4^D,D,4]]\) error-correcting family [20]. Various other concatenations give families with increasing distance (see cousins). |

\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code | Member of a family of self-dual CCS codes constructed from \([2^r-1,2^r-r-1,3]=C_X=C_Z\) Hamming codes and their duals the simplex codes. The code's stabilizer generator matrix blocks \(H_{X}\) and \(H_{Z}\) are both the generator matrix for a simplex code. The weight of each stabilizer generator is \(2^{r-1}\). |

\([[2m,2m-2,2]]\) error-detecting code | CSS stabilizer 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 [21; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [22]. |

\([[4,2,2]]\) 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. |

\([[6,2,2]]\) \(C_6\) code | Error-detecting self-dual CSS code used in concatenation schemes for fault-tolerant quantum computation. A set of stabilizer generators is \(IIXXXX\) and \(XXIIXX\), together with the same two \(Z\)-type generators. |

\([[6,4,2]]\) 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 [22; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [23]. |

\([[7,1,3]]\) Steane code | A \([[7,1,3]]\) CSS code that is the smallest qubit CSS code to correct a single-qubit error [24]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors. |

\([[8,3,2]]\) CSS code | Smallest 3D color code whose physical qubits lie on vertices of a cube and which admits a (weakly) transversal CCZ gate. |

\([[9,1,3]]\) 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]
- L. Spagnoli, A. Roggero, and N. Wiebe, “Fault-tolerant simulation of Lattice Gauge Theories with gauge covariant codes”, (2024) arXiv:2405.19293
- [13]
- A. Rajput, A. Roggero, and N. Wiebe, “Quantum Error Correction with Gauge Symmetries”, (2022) arXiv:2112.05186
- [14]
- 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
- [15]
- N. Baspin, O. Fawzi, and A. Shayeghi, “A lower bound on the overhead of quantum error correction in low dimensions”, (2023) arXiv:2302.04317
- [16]
- S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
- [17]
- D. Poulin, J.-P. Tillich, and H. Ollivier, “Quantum serial turbo-codes”, (2009) arXiv:0712.2888
- [18]
- W. Zeng and L. P. Pryadko, “Minimal distances for certain quantum product codes and tensor products of chain complexes”, Physical Review A 102, (2020) arXiv:2007.12152 DOI
- [19]
- S. Bravyi et al., “How much entanglement is needed for quantum error correction?”, (2024) arXiv:2405.01332
- [20]
- D. Hangleiter et al., “Fault-tolerant compiling of classically hard IQP circuits on hypercubes”, (2024) arXiv:2404.19005
- [21]
- 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
- [22]
- A. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
- [23]
- H. Goto, “Many-hypercube codes: High-rate quantum error-correcting codes for high-performance fault-tolerant quantum computation”, (2024) arXiv:2403.16054
- [24]
- B. Shaw et al., “Encoding one logical qubit into six physical qubits”, Physical Review A 78, (2008) arXiv:0803.1495 DOI