# Hermitian qubit code[1]

## Description

An \([[n,k,d]]\) stabilizer code constructed from a Hermitian self-orthogonal linear quaternary 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 \(GF(4)\).

Hermitian codes are in one-to-one correspondence with trace-Hermitian self-orthogonal additive codes via the \(GF(4)\) representation. 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.

All of its automorphisms lie in the Clifford group [4; Corr. 16].

## Protection

A Hermitian self-orthogonal linear \([n,k,d]_{4}\) code yields an \([[n,n-2k]]\) qubit stabilizer code with distance no less than \(d\); this is the qubit Hermitian construction. A variant construction allows for the use of nearly self-orthogonal codes [5].

The stabilizer generator matrix is of the form \begin{align} H=\begin{pmatrix}H\\ \alpha H \end{pmatrix}~, \tag*{(1)}\end{align} where \(H\) is the parity-check matrix of the classical code.

## Transversal Gates

## Fault Tolerance

## Notes

## Parents

## Children

- \([[13,1,5]]\) cyclic code — The \([[13,1,5]]\) cyclic code is derived from a quadratic residue code using the Hermitian construction [10]; see [11; pg. 11] for details.
- Five-qubit perfect code — The five-qubit code is derived from the \([5,3,3]_4\) shortened hexacode via the qubit Hermitian construction [12][13; Exam. A].
- Camara-Ollivier-Tillich code

## Cousins

- Dual linear code — Hermitian qubit codes are constructed from Hermitian self-orthogonal linear codes over \(GF(4)\) via the \(GF(4)\) representation.
- Constacyclic code — Duadic constacyclic codes yield many examples of Hermitian qubit codes [14].
- Graph-adjacency code — Bounds on self-dual \([[n,0,d]]\) Hermitian codes based on graphs have been derived [9].
- Perfect-tensor code — The sole codeword of some \([[n,0,d]]\) Hermitian codes is an AME state [15].
- Perfect quantum code — The only perfect qubit codes are the Hermitian qubit code family \([[(4^r-1)/3, (4^r-1)/3 - 2r, 3]]\) for \(r \geq 2\), obtained from Hamming codes over \(GF(4)\) [1,16].
- Qubit BCH code — Hermitian self-orthogonal quaternary BCH codes are used to construct a subset of qubit BCH codes via the Hermitian 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, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
- [5]
- P. Lisoněk and V. Singh, “Quantum codes from nearly self-orthogonal quaternary linear codes”, Designs, Codes and Cryptography 73, 417 (2014) DOI
- [6]
- E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
- [7]
- D. Gottesman, “Theory of fault-tolerant quantum computation”, Physical Review A 57, 127 (1998) arXiv:quant-ph/9702029 DOI
- [8]
- L. E. Danielsen and M. G. Parker, “On the classification of all self-dual additive codes over GF(4) of length up to 12”, Journal of Combinatorial Theory, Series A 113, 1351 (2006) arXiv:math/0504522 DOI
- [9]
- V. D. Tonchev, “Error-correcting codes from graphs”, Discrete Mathematics 257, 549 (2002) DOI
- [10]
- E. Rains, private communication, April 1997.
- [11]
- F. Vatan, V. P. Roychowdhury, and M. P. Anantram, “Spatially Correlated Qubit Errors and Burst-Correcting Quantum Codes”, (1997) arXiv:quant-ph/9704019
- [12]
- A. J. Scott, “Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions”, Physical Review A 69, (2004) arXiv:quant-ph/0310137 DOI
- [13]
- G. D. Forney, M. Grassl, and S. Guha, “Convolutional and Tail-Biting Quantum Error-Correcting Codes”, IEEE Transactions on Information Theory 53, 865 (2007) arXiv:quant-ph/0511016 DOI
- [14]
- R. Dastbasteh et al., “An infinite class of quantum codes derived from duadic constacyclic codes”, (2024) arXiv:2312.06504
- [15]
- Z. Raissi, “Modifying Method of Constructing Quantum Codes From Highly Entangled States”, IEEE Access 8, 222439 (2020) arXiv:2005.01426 DOI
- [16]
- D. Gottesman, “Pasting Quantum Codes”, (1996) arXiv:quant-ph/9607027

## Page edit log

- Victor V. Albert (2024-06-24) — most recent
- Simon Burton (2024-06-24)
- Victor V. Albert (2022-07-21)
- Marianna Podzorova (2021-12-13)

## Cite as:

“Hermitian qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/stabilizer_over_gf4