## Description

## Protection

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~, \tag*{(1)}\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}}~. \tag*{(2)}\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 [1–3]. 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.

## Decoding

## Notes

## Parents

- Block quantum code
- Finite-dimensional quantum error-correcting code
- Group-based quantum 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]. Interpreted this way, Galois-qudit codes are group quantum codes whose physical spaces are constructed using Galois fields \(GF(q)\) as groups.
- Category-based quantum code — Category quantum codes whose physical spaces are constructed using \(GF(q)\) as the category are Galois-qudit codes.

## Children

- Qubit code — Galois-qudit quantum codes for \(q=2\) correspond to qubit codes.
- Galois-qudit non-stabilizer code

## Cousins

- 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.
- Subsystem Galois-qudit code — Subsystem Galois-qudit codes reduce to (subspace) Galois-qudit codes when there is no gauge subsystem.

## References

- [1]
- E. Knill, “Non-binary Unitary Error Bases and Quantum Codes”, (1996) arXiv:quant-ph/9608048
- [2]
- E. Knill, “Group Representations, Error Bases and Quantum Codes”, (1996) arXiv:quant-ph/9608049
- [3]
- A. Klappenecker and M. Roetteler, “Beyond Stabilizer Codes I: Nice Error Bases”, (2001) arXiv:quant-ph/0010082
- [4]
- K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, “Discrete phase space based on finite fields”, Physical Review A 70, (2004) arXiv:quant-ph/0401155 DOI
- [5]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [6]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074

## Page edit log

- Victor V. Albert (2022-05-07) — most recent
- Victor V. Albert (2021-12-03)

## Cite as:

“Galois-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_into_galois