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

## Description

An \([[n,k,d]]\) stabilizer code constructed from a quaternary classical code using the one-to-one correspondence between the four Pauli matrices \(\{I,X,Y,Z\}\) and the four elements \(\{0,1,\alpha^2,\alpha\}\) of the quaternary field \(GF(4)\).

An \(n\)-qubit Pauli stabilizer can be represented as a length-\(n\) quaternary vector. The stabilizer commutation condition corresponds to a zero trace-Hermitian inner product between the corresponding vectors. Stabilizer codes over \(GF(4)\) can thus be constructed from classical trace-Hermitian self-orthogonal additive quaternary codes and Hermitian self-orthogonal linear quaternary codes (since the latter are automatically trace-Hermitian self-orthogonal). The classical code corresponds to the stabilizer group \(\mathsf{S}\) while its trace-Hermitian dual corresponds to the normalizer \(\mathsf{N(S)}\).

Stabilizer codes over \(GF(4)\) can be constructed as follows. Let \(\phi\) be a bijection from a linear binary subspace to \(GF(4)^n\). Let \(C\) be a trace-Hermitian self-orthogonal additive 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 \(GF(4)\).

Quaternary linear codes are Hermitian self-orthogonal (self-dual) iff they are trace-Hermitian self-orthogonal (self-dual) additive [2; Thm. 27.4.1] ([3; Thm. 9.10.3]). In other words, if the underlying quaternary code is linear, then the extra trace operation can be removed from the definition of inner product.

## Protection

## Fault Tolerance

## Notes

## Parents

- Qubit stabilizer code
- Stabilizer code over \(GF(q^2)\) — Stabilizer codes over \(GF(q^2)\) for \(q=2\) are stabilizer codes over \(GF(4)\).

## Child

- Five-qubit perfect code — The five-qubit code is derived from the \([5,3,3]_4\) shortened hexacode via the stabilizer-over-\(GF(4)\) construction.

## Cousins

- Dual additive code — The stabilizer commutation condition for stabilizer codes over \(GF(4)\) can equivalently be stated in the representation of stabilizers as quaternary vectors. A pair of \(n\)-qubit stabilizers commute iff the trace-Hermitian inner product of their corresponding vectors is zero. Stabilizer codes over \(GF(4)\) can thus be constructed from trace-Hermitian self-orthogonal additive quaternary codes.
- Qubit BCH code — Hermitian self-orthogonal quaternary BCH codes are used to construct a subset of qubit BCH codes via the stabilizer-over-\(GF(4)\) construction.

## References

- [1]
- A. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
- [2]
- M. F. Ezerman, "Quantum Error-Control Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [3]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [4]
- E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048

## Page edit log

- Victor V. Albert (2022-07-21) — most recent
- Marianna Podzorova (2021-12-13)

## 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