Galois-qudit stabilizer code[1,2] 


An \(((n,K,d))_q\) Galois-qudit code whose logical subspace is the joint eigenspace of commuting Galois-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 Galois-qudit Pauli string that implements a nontrivial logical operation in the code.

A Galois-qudit stabilizer code encoding an integer number of qudits (\(K=q^k\)) is denoted as \([[n,k]]_q\) or \([[n,k,d]]_q\). This notation differentiates between Galois-qudit and modular-qudit \([[n,k,d]]_{\mathbb{Z}_q}\) stabilizer codes, although the same notation is usually used for both. Galois-qudit stabilizer codes need not encode an integer number of qudits, with \(K=q^{n-\frac{r}{m}}\), where \(r\) is the number of generators of the stabilizer group, and \(q=p^m\) given prime \(p\) for all Galois qudits. As a result, \([[n,k,d]]\) notation is often used with non-integer \(k=\log_q K\).

Galois symplectic representation: The single Galois-qudit Pauli string \(X_{a} Z_{b}\) for \(a,b\in GF(q)\) is converted to the vector \(a|b\). The multi Galois-qudit version follows naturally.

A pair of Galois-qudit stabilizers on \(n\) Galois qudits with symplectic representation vectors \((a|b)\) and \((a^{\prime}|b^{\prime})\) commute iff their trace symplectic inner product is zero, \begin{align} \text{tr}(a \cdot b^{\prime} - a^{\prime}\cdot b) = \sum_{j=1}^{n} \text{tr}(a_j b^{\prime}_j - a^{\prime}_i b_i) = 0~. \tag*{(1)}\end{align} Symplectic representations of stabilizer group elements form a trace-symplectic self-orthogonal linear code over \(GF(q)^{2n}\). The trace-symplectic inner product reduces to the symplectic inner product when the trace is removed, and a symplectic self-orthogonal set of vectors is automatically trace-symplectic self-orthogonal.

Another correspondence between Galos-qudit Pauli matrices and elements of the Galois field \(GF(q^2)\) yields the one-to-one correspondence between Galois-qudit stabilizer codes and trace-alternating self-orthogonal additive codes over \(GF(q^2)\) [2].

\(GF(q^2)\) representation: An \(n\)-qubit Galois-qudit Pauli stabilizer can be represented as a length-\(n\) vector over \(GF(q^2)\) using the one-to-one correspondence between the \(q^2\) Galois-qudit Pauli matrices and elements of \(GF(q^2)\).

The sets of \(GF(q^2)\)-represented vectors for all generators yield a trace-alternating self-orthogonal additive code over \(GF(q^2)\).

Galois-qudit stabilizer codes can equivalently [3] (see also [4]) be defined using graphs, yielding an analytical form for the codewords [5].


Detects errors on up to \(d-1\) qudits, and corrects erasure errors on up to \(d-1\) qudits. Corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) qudits.


Encoder with \(O(n^2)\) gates can be determined in classical runtime of order \(O(n^3)\) [6].


As opposed to modular qudits for composite \(q\), Galois qudits inherit most of the properties of the prime-qudit Clifford group due to the correspondence between a \(q=p^m\) Galois qudit and \(m\) prime qudits of dimension \(p\) [1].


The number of Galois-qudit stabilizer codes was determined in Ref. [7].See Quantum Codes qudit stabilizer database, maintained by N. Aydin, P. Liu, and B. Yoshino, at this website.


  • Galois-qudit USt code — A Galois-qudit stabilizer code with stabilizer group \(\mathsf{S}\) can be thought of as a Galois-qudit USt with only the identity coset representative. Conversely, if \(K = q^k\), and if the set of coset representatives of a Galois-qudit USt form a \(q\)-ary linear code, then they can be absorbed into a Galois-qudit stabilizer group that defines the USt.
  • Stabilizer code



  • Galois-qudit CWS code — Galois-qudit CWS codes whose underlying classical code is a linear \(q\)-ary code are Galois-qudit stabilizer codes containing a cluster-state codeword.
  • Modular-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 \(k=nm-r\) [1]. The case \(m=1\) reduces to conventional prime-qudit stabilizer codes on \(n\) qudits.
  • Additive \(q\)-ary code — Galois-qudit stabilizer codes are the closest quantum analogues of additive codes over \(GF(q)\) because addition in the field corresponds to multiplication of stabilizers in the quantum case.
  • Dual additive code — Galois-qudit stabilizer codes are in one-to-one correspondence with trace-symplectic self-orthogonal additive codes of length \(2n\) over \(GF(q)\) via the Galois symplectic representation [1]. They are also in one-to-one correspondence with trace-alternating self-orthogonal additive codes of length \(n\) over \(GF(q^2)\) via the \(GF(q^2)\) representation.
  • Subsystem Galois-qudit stabilizer code — Subsystem Galois-qudit stabilizer codes reduce to Galois-qudit stabilizer codes when there are no gauge qudits.


A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
A. Ketkar et al., “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
M. Grassl, A. Klappenecker, and M. Rotteler, “Graphs, quadratic forms, and quantum codes”, Proceedings IEEE International Symposium on Information Theory, arXiv:quant-ph/0703112 DOI
D. Schlingemann and R. F. Werner, “Quantum error-correcting codes associated with graphs”, Physical Review A 65, (2001) arXiv:quant-ph/0012111 DOI
M. GRASSL, M. RÖTTELER, and T. BETH, “EFFICIENT QUANTUM CIRCUITS FOR NON-QUBIT QUANTUM ERROR-CORRECTING CODES”, International Journal of Foundations of Computer Science 14, 757 (2003) arXiv:quant-ph/0211014 DOI
D. Gross, “Hudson’s theorem for finite-dimensional quantum systems”, Journal of Mathematical Physics 47, (2006) arXiv:quant-ph/0602001 DOI
Page edit log

Your contribution is welcome!

on (edit & pull request)— see instructions

edit on this site

Zoo Code ID: galois_stabilizer

Cite as:
“Galois-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_galois_stabilizer, title={Galois-qudit stabilizer code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:

Cite as:

“Galois-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.