Stabilizer code over \(GF(4)\)[1]

Description

An \([[n,k,d]]\) stabilizer code whose encoding is based on a weakly self-dual additive quaternary code \((n, 2^{n-k}, d^*)_4\) with respect to the trace inner product where \(d \ge d^*\). The quaternary field \(GF(4)=\mathbf{F}_4\) consists of \(\{0, 1, w, \bar{w}\}\), with \(\bar{w} = w^2 = w + 1\), \(\mathrm{Tr}(x) = x+\bar{x}\), and trace inner product \(u * v = \mathrm{Tr}(u \cdot \bar{v})\). There is a mapping \(L\) between Pauli matrices \(I, Y, Z, X\) and \(0, 1, \bar{w}, w\), in turn \([A, B] \Leftrightarrow Tr\langle L(A), L(A)\rangle\). The classical self-dual code \(C\) over \(GF(4)^n\) corresponds to the stabilizer group \(\mathsf{S}\) while \(C^{\perp}\) corresponds to \(\mathsf{N(S)}\).

The quaternary code needs to only be additive (its codewords are closed under addition). It need not be a linear code, which would require the set of codewords to be closed under multiplication as well.

Protection

Detects errors on \(d-1\) qubits, corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) qubits.

Notes

Tables of \([[n,0,d]]\) codes, corresponding to a self-dual \(GF(4)\) representation, at this website.

Parent

Child

Cousins

  • Additive \(q\)-ary code — Let \(\phi\) be a bijection from a linear binary subspace to \(GF(4)^n\). Let \(C\) be an additive self-orthogonal subcode over \(GF(4)\), containing \(2^{n-k}\) vectors, such that there are no vectors of weight less than \(d\) in \(C^{\perp}\setminus C\). Then, any eigenspace of the inverse map \(\phi^{-1}(C)\) is an \([[n, k, d]]\) stabilizer code over \(GF(4)\).
  • Stabilizer code over \(GF(q^2)\) — Stabilizer codes over \(GF(q^2)\) are Galois-qudit extensions of those over \(GF(4)\).

Zoo code information

Internal code ID: stabilizer_over_gf4

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: stabilizer_over_gf4

Cite as:
“Stabilizer code over \(GF(4)\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stabilizer_over_gf4
BibTeX:
@incollection{eczoo_stabilizer_over_gf4, title={Stabilizer code over \(GF(4)\)}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stabilizer_over_gf4} }
Permanent link:
https://errorcorrectionzoo.org/c/stabilizer_over_gf4

References

[1]
A. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”. quant-ph/9608006

Cite as:

“Stabilizer code over \(GF(4)\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stabilizer_over_gf4

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/stabilizer_over_gf4.yml.