Modular-qudit stabilizer code[1]


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.

A modular-qudit stabilizer code encoding an integer number of qudits (\(K=q^k\)) is denoted as \([[n,k]]_{q}\) or \([[n,k,d]]_{q}\). For composite \(q\), such codes need not encode an integer number of qudits, with \(K=q^n/|\mathsf{S}|\) [2]. This is because \(|{\mathsf{S}}|\) need not be a power of \(q\), as group generators may have different orders. As a result, \([[n,k,d]]\) notation is often used with non-integer \(k=\log_q K\). Prime-qudit stabilizer codes, where \(q=p\) for some prime \(p\), do not suffer from this issue and encode \(n-k\) logical qudits, with \(K=p^{n-k}\).

Each code can be represented by a check matrix (a.k.a. stabilizer generator matrix) \(H=(A|B)\), where each row \((a|b)\) is the \(q\)-ary symplectic representation of a stabilizer generator. The check matrix can be brought into standard form via Gaussian elimination [2].


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 \(E,F \in {\mathcal{E}}\).


The structure of stabilizer codes allows for syndrome-based decoding, where errors are corrected based on the results of stabilizer measurements (syndromes).Trellis decoder for prime-dimensional qudits, which builds a compact representation of the algebraic structure of the normalizer \(\mathsf{N(S)}\) [3].


A standardized definition of the qudit stabilizer group is developed in [2].The number of modular-qudit stabilizer codes was determined in Ref. [4].


  • Stabilizer code
  • Modular-qudit code
  • Quantum Lego code — Modular-qudit stabilizer codes are quantum Lego codes built out of atomic blocks such as the 2-qudit repetition code, single-qudit trivial stabilizer codes, and tensor-products of the \(|0\rangle\) state.



  • Abelian topological code — All abelian topological orders can be realized as modular-qudit stabilizer codes [5].
  • Galois-qudit stabilizer code — Recalling that \(q=p^m\), Galois-qudit stabilizer codes can also be treated as prime-qudit stabilizer codes on \(mn\) qudits, giving \(k=nm-r\) [6]. The case \(m=1\) reduces to conventional prime-qudit stabilizer codes on \(n\) qudits.
  • Qubit stabilizer code — Modular-qudit stabilizer codes for prime-dimensional qudits \(q=p\) inherit most of the features of qubit stabilizer codes, including encoding an integer number of qudits and a Pauli group with a unique number of generators. Conversely, qubit codes can be extended to modular-qudit codes by decorating appropriate generators with powers. For example, \([[4,2,2]]\) qubit code generators can be adjusted to \(ZZZZ\) and \(XX^{-1} XX^{-1}\). A systematic procedure extending a qubit code to prime-qudit codes involves putting its generator matrix into local-dimension-invariant (LDI) form [7]. Various bounds exist on the distance of the resulting codes [8][9].
  • Translationally-invariant stabilizer code — Modular-qudit stabilizer codes can be thought of as translationally-invariant stabilizer codes for dimension \(D = 0\), with the lattice consisting of a single site.


Daniel Gottesman, “Stabilizer Codes and Quantum Error Correction”. quant-ph/9705052
V. Gheorghiu, “Standard form of qudit stabilizer groups”, Physics Letters A 378, 505 (2014). DOI; 1101.1519
Eric Sabo, Arun B. Aloshious, and Kenneth R. Brown, “Trellis Decoding For Qudit Stabilizer Codes And Its Application To Qubit Topological Codes”. 2106.08251
Tanmay Singal et al., “Counting stabiliser codes for arbitrary dimension”. 2209.01449
T. D. Ellison et al., “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022). DOI; 2112.11394
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001). DOI
L. G. Gunderman, “Local-dimension-invariant qudit stabilizer codes”, Physical Review A 101, (2020). DOI; 1910.08122
Arun J. Moorthy and Lane G. Gunderman, “Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise”. 2110.11510
L. G. Gunderman, “Degenerate local-dimension-invariant stabilizer codes and an alternative bound for the distance preservation condition”, Physical Review A 105, (2022). DOI; 2110.15274
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Internal code ID: qudit_stabilizer

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“Modular-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_qudit_stabilizer, title={Modular-qudit stabilizer code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Modular-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.