## Description

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 \(((n,K))_q\) or \(((n,K,d))_q\), whenever the code''s distance \(d\) is defined. This notation differentiates between Galois-qudit and \(((n,K,d))_{\mathbb{Z}_q}\) modular-qudit codes, although the same notation is usually used for both.'

## Protection

### Galois-qudit Pauli-string error basis

A convenient and often considered error set is the Galois-qudit analogue of the Pauli string set for qubit codes.

Galois-qudit Pauli strings: 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 \(\text{tr}\) is the field trace. For multiple Galois qudits, error set elements are tensor products of elements of the single-qudit error set. Tensor products of \(X\) (\(Z\)) Galois-qudit Paulis acting on different qudits are called \(X\)-type (\(Z\)-type) Galois-qudit Pauli strings. Combining the \(X\)-type and \(Z\)-type strings with a \(p\)th root of unity forms a group called the Galois-qudit Pauli group (on \(n\) Galois qudits.

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. 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.

## Gates

## Decoding

## Notes

## Parents

- Block quantum code
- Group-based quantum code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [3]; see Sec. 5.3 of Ref. [4]. 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 the group \(GF(q)\) as the category are Galois-qudit codes.

## Children

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

## Cousins

- Modular-qudit code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits; see [3,5][4; Sec. 5.3]. 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.
- EA Galois-qudit code — EA Galois-qudit codes utilize additional ancillary Galois qudits in a pre-shared entangled state, but reduce to ordinary Galois-qudit codes when said qudits are interpreted as noiseless physical qudits.
- Subsystem Galois-qudit code — Subsystem Galois-qudit codes reduce to (subspace) Galois-qudit codes when there is no gauge subsystem.

## References

- [1]
- 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
- [2]
- 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
- [3]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [4]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
- [5]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL

## 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