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. 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.
Cousins:
Approximate secret-sharing 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:
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, Translationally-invariant stabilizer code.

Frobenius code[3]
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.

Double-semion code[4]
Stub.
Parents:
Modular-qudit stabilizer code, Abelian topological code.
Cousins:
Kitaev surface code.

Modular-qudit CSS code[5][6]
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, Prime-qudit polynomial code (QPyC), 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.

Modular-qudit surface code[7]
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.
Protection: When defined on an \(L\times L\) square tiling of the torus, protects against \(L\) errors. More generally, the code distance is the number of edges in the shortest non contractible cycle in the tesselation or dual tesselation [8].
Parents:
Modular-qudit CSS code, Abelian topological code.
Cousins:
Kitaev surface code.
Cousin of:
Quantum-double code, Translationally-invariant stabilizer code.

Prime-qudit polynomial code (QPyC)[9]
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}
Parents:
Modular-qudit CSS code.
Cousins:
Reed-Solomon (RS) code, Cyclic code, Quantum maximum-distance-separable (MDS) code.
Cousin of:
Galois-qudit polynomial code (QPyC).

\([[2^r-1, 2^r-2r-1, 3]]_p\) prime-qudit CSS code[10]
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 [10].
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]
- Sagarmoy Dutta and Piyush P Kurur, “Quantum Cyclic Code of length dividing $p^{t}+1$”. 1011.5814
- [4]
- 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
- [5]
- Alexei Ashikhmin and Emanuel Knill, “Nonbinary Quantum Stabilizer Codes”. quant-ph/0005008
- [6]
- Avanti Ketkar et al., “Nonbinary stabilizer codes over finite fields”. quant-ph/0508070
- [7]
- A. Y. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003). DOI; quant-ph/9707021
- [8]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002). DOI; quant-ph/0110143
- [9]
- 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
- [10]
- Arun J. Moorthy and Lane G. Gunderman, “Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise”. 2110.11510