Galois-qudit code[1][2]


Also called a \(GF(q)\)- or \(\mathbb{F}_q\)-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 Galois field \(GF(q)\) and with \(q\) being a power of a prime \(p\). A Galois field can be thought of as a vector space whose basis vectors are the \(m\) roots of some polynomial and whose coefficients (i.e., field) are \(p\)th roots of unity. Codes can be denoted as \(((n,K))_{GF(q)}\) or \(((n,K,d))_{GF(q)}\), whenever the code's distance \(d\) is defined. This notation differentiates between Galois-qudit and modular-qudit codes, although the same notation, \(((n,K,d))_q\), is usually used for both.


A convenient and often considered error set is the Galois-qudit analogue of the Pauli string set for qubit codes. For a single Galois qudit, this set consists of products of \(X\)-type and \(Z\)-type operators labeled by elements \(\beta \in GF(q)\), which act on computational basis states \(|\gamma\rangle\) for \(\gamma\in GF(q)\) as \begin{align} X_{\beta}\left|\gamma\right\rangle =\left|\gamma+\beta\right\rangle \,\,\text{ and }\,\,Z_{\beta}\left|\gamma\right\rangle =e^{i\frac{2\pi}{p}\text{tr}(\beta\gamma)}\left|\gamma\right\rangle~, \end{align} where the trace maps elements of the field to elements of \(\mathbb{Z}_p\) as \begin{align} \text{tr}(\gamma)=\sum_{k=0}^{m-1}\gamma^{p^{k}}~. \end{align} For multiple Galois qudits, error set elements are tensor products of elements of the single-qudit error set.

The Galois-qudit Pauli error set is a unitary basis for linear operators on the multi-qudit Hilbert space that is orthonormal under the Hilbert-Schmidt inner product; it is a nice error basis [3]. The distance associated with this set is often the minimum weight of a Galois qudit Pauli string that implements a nontrivial logical operation in the code.


For few-qudit codes (\(n\) is small), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the maximum-likelihood decoder.


Introduction to Galois qudits by Gottesman.Wigner function for Galois qudits [4].




  • Modular-qudit code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [5]; see Sec. 5.3 of Ref. [6]. The two coincide when \(q\) is prime, and reduce to qubits when \(q=2\). However, Pauli matrices for the two types of qudits are defined differently.


J. Bierbrauer and Y. Edel, “Quantum twisted codes”, Journal of Combinatorial Designs 8, 174 (2000). DOI
Avanti Ketkar et al., “Nonbinary stabilizer codes over finite fields”. quant-ph/0508070
E. Knill, “Non-binary Unitary Error Bases and Quantum Codes”. quant-ph/9608048
K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, “Discrete phase space based on finite fields”, Physical Review A 70, (2004). DOI; quant-ph/0401155
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001). DOI
Annika Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”. quant-ph/0501074
Page edit log

Zoo code information

Internal code ID: galois_into_galois

Your contribution is welcome!

on (edit & pull request)

edit on this site

Zoo Code ID: galois_into_galois

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

Cite as:

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