Galois-qudit code 

Also known as \(GF(q)\)-qudit code, \(\mathbb{F}_q\)-qudit code, Galois-qudit subspace code.
Root code for the Galois-qudit Kingdom

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 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. 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

The normalizer of the Galois-qudit Pauli group is the Galois-qudit Clifford group [1].

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 (ML) decoder.

Notes

Introduction to Galois qudits by Gottesman.Wigner function for Galois qudits [2].

Parents

Children

Cousins

  • Modular-qudit 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]. 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]
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
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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} }
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“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/edit/main/codes/quantum/qudits_galois/galois_into_galois.yml.