Galois-qudit stabilizer code[1][2]

Description

An \(((n,K,d))_{GF(q)}\) Galois-qudit code whose logical subspace is the joint eigenspace of commuting Galois-qudit Pauli operators forming the code's stabilizer group \(\mathsf{S}\). Traditionally, the logical subspace is the joint \(+1\) eigenspace, and the stabilizer group does not contain \(e^{i \phi} I\) for any \(\phi \neq 0\). The distance \(d\) is the minimum weight of a Galois-qudit Pauli string that implements a nontrivial logical operation in the code.

A Galois-qudit stabilizer code encoding an integer number of qudits (\(K=q^k\)) is denoted as \([[n,k]]_{GF(q)}\) or \([[n,k,d]]_{GF(q)}\). This notation differentiates between Galois-qudit and modular-qudit stabilizer codes, although the same notation, \([[n,k,d]]_q\), is usually used for both. Galois-qudit stabilizer codes need not encode an integer number of qudits, with \(K=q^{n-\frac{r}{m}}\), where \(r\) is the number of generators of the stabilizer group, and \(q=p^m\) given prime \(p\) for all Galois qudits. As a result, \([[n,k,d]]\) notation is often used with non-integer \(k=\log_q K\).

Protection

Detects errors on up to \(d-1\) qudits, and corrects erasure errors on up to \(d-1\) qudits. Corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) qudits.

Gates

As opposed to modular qudits for composite \(q\), Galois qudits inherit most of the properties of the prime-qudit Clifford group due to the correspondence between a \(q=p^m\) Galois qudit and \(m\) prime qudits of dimension \(p\) [1].

Parents

  • Stabilizer code
  • Galois-qudit non-stabilizer code — A non-stabilizer code is also a stabilizer code if its Fourier description \(\mathsf{B}\) is a subgroup of some Gottesman subgroup \(\mathsf{S}\). When \(\mathsf{B}\) is just a subset, the code is explicitly not a stabilizer code.

Children

Cousins

  • Modular-qudit stabilizer code — Recalling that \(q=p^m\), Galois-qudit stabilizer codes can also be treated as prime-qudit stabilizer codes on \(mn\) qudits, giving \(k=nm-r\) [1]. The case \(m=1\) reduces to conventional prime-qudit stabilizer codes on \(n\) qudits.
  • Additive \(q\)-ary code — A Galois-qudit stabilizer code is the closest quantum analogue of an additive code over \(GF(q)\) because addition in the field corresponds to multiplication of stabilizers in the quantum case.

Zoo code information

Internal code ID: galois_stabilizer

Your contribution is welcome!

on github.com (edit & pull request)

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Zoo Code ID: galois_stabilizer

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

References

[1]
Alexei Ashikhmin and Emanuel Knill, “Nonbinary Quantum Stabilizer Codes”. quant-ph/0005008
[2]
Avanti Ketkar et al., “Nonbinary stabilizer codes over finite fields”. quant-ph/0508070

Cite as:

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

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