Galois-qudit code[1][2]

Description

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.

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~, \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.

Decoding

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.

Notes

Introduction to Galois qudits by Gottesman.

Parent

Child

Cousin

  • Modular-qudit code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [4]; see Sec. 5.3 of Ref. [5]. 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.

Zoo code information

Internal code ID: galois_into_galois

Your contribution is welcome!

on github.com (edit & pull request)

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Zoo Code ID: galois_into_galois

Cite as:
“Galois-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_into_galois
BibTeX:
@incollection{eczoo_galois_into_galois, title={Galois-qudit code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/galois_into_galois} }
Permanent link:
https://errorcorrectionzoo.org/c/galois_into_galois

References

[1]
J. Bierbrauer and Y. Edel, “Quantum twisted codes”, Journal of Combinatorial Designs 8, 174 (2000). DOI
[2]
Avanti Ketkar et al., “Nonbinary stabilizer codes over finite fields”. quant-ph/0508070
[3]
E. Knill, “Non-binary Unitary Error Bases and Quantum Codes”. quant-ph/9608048
[4]
Alexei Ashikhmin and Emanuel Knill, “Nonbinary Quantum Stabilizer Codes”. quant-ph/0005008
[5]
Annika Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”. quant-ph/0501074

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qudits_galois/galois_into_galois.yml.