Welcome to the Modular-qudit Kingdom.

Three qutrit code[1]
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. It is also the smallest non-trivial instance of a quantum maximum distance separable code (QMDS), saturating the quantum Singleton bound. 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.
Protection: Detects single qutrit errors and protects against a single-qutrit erasure. There does not exist a three-qubit code with analogous properties.
Parents:
Modular-qudit CSS code, Holographic code, Quantum maximum-distance-separable (MDS) code.
Cousins:
Approximate secret-sharing code.
Cousin of:
\([[3,1,2]]_{\mathbb Z}\) rotor code.

Modular-qudit code
Also called a \(\mathbb{Z}_q\)-qudit code. Encodes \(K\)-dimensional Hilbert space into a \(q^n\)-dimensional (\(n\)-qudit) Hilbert space, with canonical qudit states \(|k\rangle\) labeled by elements \(k\) of the group \(\mathbb{Z}_q\) of integers modulo \(q\). Usually denoted as \(((n,K))_q\) or \(((n,K,d))_q\), whenever the code's distance \(d\) is defined, and with \(q=p\) when the dimension is prime.
Protection: A convenient and often considered error set is the modular-qudit analogue of the Pauli string basis for qubit codes. For a single qudit, this set consists of products of powers of the qudit Pauli matrices \(X\) and \(Z\), which act on computational basis states \(|k\rangle\) for \(k\in\mathbb{Z}_q\) as \begin{align}
X\left|k\right\rangle =\left|k+1\right\rangle \,\,\text{ and }\,\,Z\left|k\right\rangle =e^{i\frac{2\pi}{q}k}\left|k\right\rangle ~,
\end{align} with addition performed modulo \(q\). For multiple qudits, error set elements are tensor products of elements of the single-qudit error set.
Parents:
Finite-dimensional quantum error-correcting code.
Parent of:
Modular-qudit stabilizer code.
Cousin of:
Entanglement-assisted (EA) QECC, Galois-qudit code, Group-based quantum code.

Modular-qudit stabilizer code[2]
An \(((n,K,d))_q\) modular-qudit code whose logical subspace is the joint eigenspace of commuting qudit Pauli operators forming the code's stabilizer group \(\mathsf{S}\). Traditionally, the logical subspace is the joint \(+1\) eigenspace, and the stabilizer group does not contain \(e^{i \phi} I\) for any \(\phi \neq 0\). The distance \(d\) is the minimum weight of a qudit Pauli string that implements a nontrivial logical operation in the code.
Protection: Detects errors on up to \(d-1\) qudits, and corrects erasure errors on up to \(d-1\) qudits. More generally, define the normalizer \(\mathsf{N(S)}\) of \(\mathsf{S}\) to be the set of all operators that commute with all \(S\in\mathsf{S}\). A stabilizer code can correct a Pauli error set \({\mathcal{E}}\) if and only if \(E^\dagger F \notin \mathsf{N(S)}\setminus \mathsf{S}\) for all \(E,F \in {\mathcal{E}}\).
Parents:
Stabilizer code, Modular-qudit code, Quantum Lego code.
Parent of:
Double-semion code, Frobenius code, Modular-qudit CSS code.
Cousin of:
Abelian topological code, Galois-qudit stabilizer code, Qubit stabilizer code, Subsystem modular-qudit stabilizer code, Translationally-invariant stabilizer code.

Double-semion code[3]
Stub.
Parents:
Modular-qudit stabilizer code, Abelian topological code.
Cousins:
Kitaev surface code.

Frobenius code[4]
Let \(C\) be a quantum cyclic code on \(n\) prime-dimensional qudits. \(C\) is a Frobenius code if there exists a positive integer \(t\) such that \(n\) divides \(p^t +1\).
Protection: Protects against Pauli noise.
Parents:
Modular-qudit stabilizer code, Quantum cyclic code.

Modular-qudit CSS code[5][6][7]
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 \(H_Z\).
Parents:
Modular-qudit stabilizer code.
Parent of:
Modular-qudit surface code, Quantum Reed-Solomon code, Three qutrit code, \([[2^r-1, 2^r-2r-1, 3]]_p\) prime-qudit CSS code.
Cousins:
Calderbank-Shor-Steane (CSS) stabilizer code, Linear \(q\)-ary code.
Cousin of:
Group GKP code.

Modular-qudit surface code[8][9][10]
Extension of the surface code to prime-dimensional [8][9] and more general modular qudits [10]. Stabilizer 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. Since qudits have more than one \(X\) and \(Z\)-type operator, various sets of stabilizer generators can be defined. Ground-state degeneracy and the associated phase depends on the qudit dimension and the stabilizer generators.
Parents:
Modular-qudit CSS code.
Cousins:
Abelian topological code, Kitaev surface code, String-net code.
Cousin of:
Quantum-double code, Translationally-invariant stabilizer code.

Quantum Reed-Solomon code[11]
Also called prime-qudit polynomial code (QPyC). Prime-qudit CSS code constructed using two Reed-Solomon codes.
Parents:
Modular-qudit CSS code.
Cousins:
Reed-Solomon (RS) code, Quantum maximum-distance-separable (MDS) code.
Cousin of:
Galois-qudit RS code.

\([[2^r-1, 2^r-2r-1, 3]]_p\) prime-qudit CSS code[12]
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 [12].
Parents:
Modular-qudit CSS code.
Cousins:
\([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code.

## References

- [1]
- R. Cleve, D. Gottesman, and H.-K. Lo, “How to Share a Quantum Secret”, Physical Review Letters 83, 648 (1999). DOI; quant-ph/9901025
- [2]
- Daniel Gottesman, “Stabilizer Codes and Quantum Error Correction”. quant-ph/9705052
- [3]
- M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases”, Physical Review B 71, (2005). DOI; cond-mat/0404617
- [4]
- Sagarmoy Dutta and Piyush P Kurur, “Quantum Cyclic Code of length dividing $p^{t}+1$”. 1011.5814
- [5]
- A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist”, Physical Review A 54, 1098 (1996). DOI; quant-ph/9512032
- [6]
- A. M. Steane, “Error Correcting Codes in Quantum Theory”, Physical Review Letters 77, 793 (1996). DOI
- [7]
- “Multiple-particle interference and quantum error correction”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452, 2551 (1996). DOI; quant-ph/9601029
- [8]
- A. Y. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003). DOI; quant-ph/9707021
- [9]
- S. S. Bullock and G. K. Brennen, “Qudit surface codes and gauge theory with finite cyclic groups”, Journal of Physics A: Mathematical and Theoretical 40, 3481 (2007). DOI; quant-ph/0609070
- [10]
- Haruki Watanabe, Meng Cheng, and Yohei Fuji, “Ground state degeneracy on torus in a family of $\mathbb{Z}_N$ toric code”. 2211.00299
- [11]
- M. Grassl, W. Geiselmann, and T. Beth, “Quantum Reed—Solomon Codes”, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes 231 (1999). DOI; quant-ph/9910059
- [12]
- Arun J. Moorthy and Lane G. Gunderman, “Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise”. 2110.11510