Hermitian qubit code[1]
Alternative names: Calderbank-Rains-Shor-Sloane (CRSS) code, \(GF(4)\)-linear code, \(M_{3}\) code.
Description
An \([[n,k,d]]\) stabilizer code constructed from a Hermitian self-orthogonal linear quaternary code using the \(GF(4)\) representation.
Hermitian codes are in one-to-one correspondence with 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 field trace 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, related to Construction X, 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
Transversal \(SH\) and \(HS\) "facet" gates (a.k.a. \(M_3\) gates) which cyclically permute Paulis as \(X \to Y\), \(Y \to Z\), and \(Z \to X\) [6; Sec. 8.2].The three-block transversal gate mapping each physical \(X \to XYZ\) and each \(Z \to ZXY\) implements a logical gate [7][8; Exam. 2].Gates
Signed weight enumerators [9] determine performance of magic \(T\)-state distillation protocols [10].Fault Tolerance
Characterizing fault-tolerant multi-qubit gates under the \(GF(4)\) representation may involve characterizing all global automorphisms of some number of copies of a code that preserve the symplectic inner product [8; pg. 9].Notes
Tables of \([[n,0,d]]\) Hermitian codes [11,12], corresponding to a self-dual \(GF(4)\) representation, at this website. Bounds on self-dual \([[n,0,d]]\) Hermitian codes based on graphs have been derived [13].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 [13].
- Perfect-tensor code— The sole codeword of some \([[n,0,d]]\) Hermitian codes is an AME state [15].
- Self-dual linear code— Hermitian qubit codes are constructed from Hermitian self-orthogonal linear codes over \(GF(4)\) via the \(GF(4)\) representation. This relation yields bounds on self-dual codes over \(GF(4)\) [10].
- 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].
- \([[13,1,5]]\) cyclic code— A different cyclic \([[13,1,5]]\) code can be derived from a quaternary QR code using the Hermitian construction [17]; see [18; pg. 11] for details.
- Qubit BCH code— Hermitian self-orthogonal quaternary BCH codes are used to construct a subset of qubit BCH codes via the Hermitian construction.
Primary Hierarchy
Parents
Hermitian qubit code
Children
The five-qubit code is derived from the \([5,3,3]_4\) shortened hexacode via the qubit Hermitian construction [19][20; Exam. A].
References
- [1]
- A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “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]
- D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
- [7]
- D. Gottesman, “Theory of fault-tolerant quantum computation”, Physical Review A 57, 127 (1998) arXiv:quant-ph/9702029 DOI
- [8]
- E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
- [9]
- P. Rall, “Signed quantum weight enumerators characterize qubit magic state distillation”, (2017) arXiv:1702.06990
- [10]
- A. R. Kalra and S. Prakash, “Invariant Theory and Magic State Distillation”, (2025) arXiv:2501.10163
- [11]
- L. E. Danielsen, “On Self-Dual Quantum Codes, Graphs, and Boolean Functions”, (2005) arXiv:quant-ph/0503236
- [12]
- 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
- [13]
- V. D. Tonchev, “Error-correcting codes from graphs”, Discrete Mathematics 257, 549 (2002) DOI
- [14]
- R. Dastbasteh, J. E. Martinez, A. Nemec, A. deMarti iOlius, and P. C. Bofill, “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
- [17]
- E. Rains, private communication, April 1997.
- [18]
- F. Vatan, V. P. Roychowdhury, and M. P. Anantram, “Spatially Correlated Qubit Errors and Burst-Correcting Quantum Codes”, (1997) arXiv:quant-ph/9704019
- [19]
- 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
- [20]
- 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
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