Hermitian qubit code[1]
Alternative names: Calderbank-Rains-Shor-Sloane (CRSS) code, \(GF(4)\)-linear stabilizer 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 code 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. The parameters satisfy \(k \equiv n\) mod 2 [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
All code automorphisms lie in the Clifford group [4; Corr. 16], so transversal physical gates implement only Clifford logical 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].A qubit stabilizer code is Hermitian if and only if a transversal \(R\) gate leaves the stabilizer group invariant [5].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].Qubit Hermitian codes for \(n < 10\) have been classified [5].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].
- Qubit CSS code— A Hermitian qubit code that can be put into CSS form via single-qubit Clifford operations remains Hermitian [5].
- 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 [19][18; pg. 11].
- 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 \([[2m,2m-2,2]]\) error-detecting code is Hermitian [5; Table 6].
The five-qubit code is derived from the \([5,3,3]_4\) shortened hexacode via the qubit Hermitian construction [20][21; Exam. A].
The \(C_6\) code is Hermitian [5; Table 6].
The Steane code is Hermitian [5; Table 6].
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]
- A. Cross and D. Vandeth, “Small Binary Stabilizer Subsystem Codes”, (2025) arXiv:2501.17447
- [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]
- R. Dastbasteh and P. Lisonek, “New quantum codes from self-dual codes over F_4”, (2022) arXiv:2211.00891
- [20]
- 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
- [21]
- 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