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

Approximate secret-sharing code | A family of \( [[n,k,d]]_{GF(q)} \) CSS codes approximately correcting errors on up to \(\lfloor (n-1)/2 \rfloor\) qubits, i.e., with approximate distance approaching the no-cloning bound \(n/2\). Constructed using a non-degenerate CSS code, such as a polynomial quantum code, and a classical authentication scheme. The code can be viewed as an \(t\)-error tolerant secret sharing scheme. Since the code yields a small logical subspace using large registers that contain both classical and quantum information, it is not useful for practical error correction problems, but instead demonstrates the power of approximate quantum error correction. |

Binary quantum Goppa code | Also known as a quantum AG code. Binary quantum Goppa codes are a family of \( [[n,k,d]]_{GF(q)} \) CSS codes for \( q=2^m \), generated using classical Goppa codes. |

Expander lifted-product code | Family of \(G\)-lifted product codes constructed using two random classical Tanner codes defined on expander graphs. For certain parameters, this construction yields the first asymptotically good QLDPC codes [1]. |

Galois-qudit CSS code | An \([[n,k,d]]_{GF(q)} \) Galois-qudit true stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Galois-qudit Pauli strings. The stabilizer generator matrix, taking values from \(GF(q)\), is of the form \begin{align} H=\begin{pmatrix}0 & H_{Z}\\ H_{X} & 0 \end{pmatrix} \label{eq:parityg} \end{align} such that the rows of the two blocks must be orthogonal \begin{align} H_X H_Z^T=0~. \label{eq:commG} \end{align} The above condition guarantees that the \(X\)-stabilizer generators, defined in the symplectic representation as rows of \(H_X\), commute with the \(Z\)-stabilizer generators associated with \(H_Z\). |

Galois-qudit polynomial code (QPyC) | Also called quantum Reed-Solomon code. An \([[n,k,n-k+1]]_{GF(q)}\) (with \(q>n\)) Galois-qudit CSS code constructed using two Reed-Solomon codes over \(GF(q)\). Let \(C_1\) be a \([n,k_1,d_1]_q\) Reed-Solomon code and \(C_2^\perp\) be a \([n,k_2,d_2]_q\) Reed-Solomon code, modified such that \(C_2^\perp \subseteq C_1\) and \(0\le k_2 \le k_1 \le n\). Then, a polynomial code is a non-degenerate \([[n,k_2,d]]_{GF(q)}\) Galois-qudit CSS code with \(d=\min(n-k_1+1,k_1-k_2+1)\). The polynomial code is the span of the basis codewords over GF(\(q\)) \begin{align} |\overline{\beta_0,\cdots,\beta_{k_2-1}}\rangle = \sum_{(\beta_{k_2},\cdots,\beta_{k_1-1})\in GF(q) } \bigotimes_{i=1}^{n} \left|\sum_{j=0}^{k_1-1} \beta_j \alpha_i^j \right\rangle, \end{align} where \((\alpha_1, \cdots, \alpha_n)\) are \(n\) distinct points chosen for code \(C_1\) from \(GF(q)\setminus \{0\}\). |

Hypergraph product code | A family of \([[n,k,d]]\) CSS codes whose construction is based on two binary linear seed codes \(C_1\) and \(C_2\). |

Lifted-product (LP) code | Code that utilizes the notion of a lifted product in its construction. Lifted products of certain classical Tanner codes are the first (asymptotically) good QLDPC codes. |

Modular-qudit CSS code | An \(((n,K,d))_q\) modular-qudit stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Pauli strings. The stabilizer generator matrix, taking values from \(\mathbb{Z}_q\), is of the form \begin{align} H=\begin{pmatrix}0 & H_{Z}\\ H_{X} & 0 \end{pmatrix} \label{eq:parityq} \end{align} such that the rows of the two blocks must be orthogonal \begin{align} H_X H_Z^T=0~. \label{eq:commQ} \end{align} The above condition guarantees that the \(X\)-stabilizer generators, defined in the symplectic representation as rows of \(H_X\), commute with the \(Z\)-stabilizer generators associated with \(H_Z\). |

Modular-qudit surface code | A family of stabilizer codes whose generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface (with a qudit located at each edge of the tesselation). The code has \( n=E \) many physical qudits, where \( E \) is the number of edges of the tesselation, and \( k=2g \) many logical qudits, where \( g \) is the genus of the surface. |

Prime-qudit polynomial code (QPyC) | Also called quantum Reed-Solomon code. An \([[n,k,n-k+1]]_p\) (with prime \(p>n\)) prime-qudit CSS code constructed using two Reed-Solomon codes over \(GF(p)=\mathbb{Z}_p\). Let \(\{\alpha_1,\cdots,\alpha_n\}\) be \(n\) distinct nonzero elements of \(\mathbb{Z}_p\), and let \(g\) be a number satisfying \(0\leq k \leq g < n\). Then, define degree-\(g\) polynomials \begin{align} f_{\mu\cup c}\left(x\right)=\mu_{0}+\mu_{1}x+\cdots+\mu_{k-1}x^{k-1}+c_{k}x^{k}+\cdots+c_{g}x^{g}\,, \end{align} where the first \(k\) coefficients are indexed by the coefficient vector \(\mu\in\mathbb{Z}_p^{\times k}\), and the remaining coefficients are indexed by the vector \(c\in\mathbb{Z}_p^{\times (g+1-k)}\). Logical states, labeled by \(\mu\), are superpositions of canonical basis states whose \(i\)th bit is \(f_{\mu\cup c}\), evaluated at \(\alpha_i\) and summed over all possible vectors \(c\), \begin{align} |\overline{\mu}\rangle=\sum_{c\in\mathbb{Z}_{p}^{\times(g+1-k)}}|f_{\mu\cup c}(\alpha_{1}),|f_{\mu\cup c}(\alpha_{2}),\cdots,|f_{\mu\cup c}(\alpha_{n})\rangle. \end{align} |

Quantum expander code | CSS codes constructed from a hypergraph product of bipartite expander graphs with bounded left and right vertex degrees. For every bipartite graph there is an associated matrix (the parity check matrix) with columns indexed by the left vertices, rows indexed by the right vertices, and 1 entries whenever a left and right vertex are connected. This matrix can serve as the parity check matrix of a classical code. Two bipartite expander graphs can be used to construct a quantum CSS code (the quantum expander code) by using the parity check matrix of one as \(X\) checks, and the parity check matrix of the other as \(Z\) checks. |

Skew-cyclic CSS code | Stub. |

Three qutrit code | A \([[3,1,2]]_3\) prime-qudit CSS code with stabilizer generators \(ZZZ\) and \(XXX\). The code defines a quantum secret-sharing scheme and serves as a minimal model for the AdS/CFT holographic duality. The codewords are \begin{align} \begin{split} | \overline{0} \rangle &= \frac{1}{\sqrt{3}} (| 000 \rangle + | 111 \rangle + | 222 \rangle) \\ | \overline{1} \rangle &= \frac{1}{\sqrt{3}} (| 012 \rangle + | 120 \rangle + | 201 \rangle) \\ | \overline{2} \rangle &= \frac{1}{\sqrt{3}} (| 021 \rangle + | 102 \rangle + | 210 \rangle)~. \end{split} \end{align} The elements in the superposition of each logical codeword are related to each other via cyclic permutations. |

\([[2^r-1, 2^r-2r-1, 3]]_p\) prime-qudit CSS code | A family of CSS codes extending Hamming-based CSS codes to prime qudits of dimension \(p\) by expressing the qubit code stabilizers in local-dimension-invariant (LDI) form [2]. |

## References

- [1]
- Pavel Panteleev and Gleb Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”. 2111.03654
- [2]
- Arun J. Moorthy and Lane G. Gunderman, “Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise”. 2110.11510