Alternative names: Weakly self-dual CSS code, Symmetric CSS code, Self-orthogonal CSS code, Homogeneous CSS code.
Description
A qubit CSS code for which transversal Hadamard is a logical operation. Equivalently, the stabilizer group is preserved by exchanging \(X\)-type and \(Z\)-type Pauli operators.
Specializing the CSS construction to the case when \(C_Z=[n,k]\) is dual-containing and \(C_X=C_Z\) yields an \([[n,2k-n]]\) self-dual qubit CSS code (a.k.a. weakly self-dual, symmetric [1], self-orthogonal, or homogeneous [2] qubit CSS code). Its \(X\)-type and \(Z\)-type stabilizers are identically supported, and transversal Hadamard preserves the stabilizer group.
Self-dual CSS codes split into normal and hyperbolic classes depending on whether transversal Hadamard acts as logical Hadamards or as pairwise logical swaps in a suitable logical basis [3; Sec. III]. Hyperbolic codes necessarily encode an even number of logical qubits and have even blocklength and distance [3; Sec. III].
Magic
Normal and hyperbolic self-dual CSS codes yield magic-state distillation protocols with asymptotically constant space overhead and yield parameter \(\gamma \to 1^{+}\) [4][3; Thms. 4.1 and 4.2].Transversal Gates
Self-dual CSS codes admit a transversal Hadamard gate. There are criteria for when such codes realize logical gates from tensor products of \(S\) and \(S^{\dagger}\) gates [5].Diagonal transversal Clifford gates on \(\ell\) codeblocks of a CSS code form \(Sp(2\ell,\mathbb{F}_2)\) for self-dual CSS codes [6].A self-dual weakly doubly even \([[n,1,d]]\) CSS code admits a partitioned transversal physical \(S\) gate that realizes \(\overline{S}^m\), where \(m=|M^+|-|M^-| \pmod 4\); for odd \(m\), together with transversal Hadamard and CNOT, this yields the full logical Clifford group transversally [7][8; Lemma 4].Fault Tolerance
Any self-dual CSS code with bounded-weight stabilizer generators admits flag fault-tolerant syndrome extraction [9].Triorthogonal codes realizing logical \(T\) gates using only physical \(T\) gates can be paired up with self-dual CSS codes to yield a transversal CNOT gate and universal fault-tolerant gates using Steane error correction [1].Cousins
- Dual linear code— Self-dual CSS codes arise from dual-containing (equivalently, self-orthogonal) binary linear codes.
- Qubit stabilizer code— Any \([[n,k,d]]\) stabilizer code can be mapped into a \([[4n,2k,2d]]\) self-dual CSS code via an intermediate tetron Majorana stabilizer code [3][10; Corr. 1], which preserves geometric locality of a code up to a constant factor.
- Tetron code— Any \([[n,k,d]]\) stabilizer code can be mapped into a \([[4n,2k,2d]]\) self-dual CSS code via an intermediate tetron Majorana stabilizer code [3][10; Corr. 1], which preserves geometric locality of a code up to a constant factor.
- Quantum Reed-Muller (RM) code— The \([[2^m,{m \choose r}, 2^{\min(r,m-r)}]]\) quantum RM family contains a self-dual sub-family for \(m=2r\), which admits logical Clifford group gates via permutations, transversal gates, and fold-transversal gates [11,12].
- Generalized quantum divisible code— Any self-dual CSS code yields a level-three generalized quantum divisible code when level-lifted [13; Thm. V.6].
- Triorthogonal code— Triorthogonal codes realizing logical \(T\) gates using only physical \(T\) gates can be paired up with self-dual CSS codes to yield a transversal CNOT gate and universal fault-tolerant gates using Steane error correction [1].
- Majorana stabilizer code— An odd-length self-dual CSS code can be converted into a complex-fermion code by replacing qubit \(Z\)-type and \(X\)-type operators with \(\gamma\)-type and \(\tilde{\gamma}\)-type Majorana operators, respectively [14].
- Color code— Color codes often have self-dual \(X\)- and \(Z\)-type bulk stabilizer structure, but boundary choices can prevent the full code from being self-dual. Thus, only color-code geometries for which transversal Hadamard is a logical operation are self-dual CSS codes.
Primary Hierarchy
Parents
Self-dual CSS codes are qubit CSS codes whose stabilizer group is preserved by transversal Hadamard.
Self-dual CSS code
Children
Puncturing a self-dual RM code yields a classical punctured RM code whose dual is its even subcode; applying the CSS construction to the even subcode yields this self-dual CSS family [15; Sec. 7.15.3].
References
- [1]
- D. Jiao, M. Bayanifar, A. Ashikhmin, and O. Tirkkonen, “Low Overhead Universal Quantum Computation with Triorthogonal Codes”, (2025) arXiv:2510.05708
- [2]
- Y.-J. Wang, Z.-Y. Xiao, Y. Zhang, X.-Y. Xiong, and S. Shi, “Construction of Multiple-Rate Quantum LDPC Codes Sharing One Scalable Stabilizer Circuit”, IEEE Transactions on Communications 71, 1071 (2023) DOI
- [3]
- J. Haah, M. B. Hastings, D. Poulin, and D. Wecker, “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
- [4]
- Quantum Information and Computation 18, (2018) arXiv:1709.02789 DOI
- [5]
- T. Tansuwannont, Y. Takada, and K. Fujii, “Clifford gates with logical transversality for self-dual CSS codes”, (2025) arXiv:2503.19790
- [6]
- S. Dasu and S. Burton, “A Classification of Transversal Clifford Gates for Qubit Stabilizer Codes”, (2025) arXiv:2507.10519
- [7]
- S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
- [8]
- S. P. Jain and V. V. Albert, “Transversal Clifford and T-Gate Codes of Short Length and High Distance”, IEEE Journal on Selected Areas in Information Theory 6, 127 (2025) arXiv:2408.12752 DOI
- [9]
- C. Chamberland and M. E. Beverland, “Flag fault-tolerant error correction with arbitrary distance codes”, Quantum 2, 53 (2018) arXiv:1708.02246 DOI
- [10]
- S. Bravyi, B. M. Terhal, and B. Leemhuis, “Majorana fermion codes”, New Journal of Physics 12, 083039 (2010) arXiv:1004.3791 DOI
- [11]
- A. Gong and J. M. Renes, “Computation with quantum Reed-Muller codes and their mapping onto 2D atom arrays”, (2024) arXiv:2410.23263
- [12]
- T. Tansuwannont, T. Chan, and R. Takagi, “Construction of the full logical Clifford group for high-rate quantum Reed-Muller codes using only transversal and fold-transversal gates”, (2026) arXiv:2602.09788
- [13]
- J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
- [14]
- A. Schuckert, E. Crane, A. V. Gorshkov, M. Hafezi, and M. J. Gullans, “Fault-tolerant fermionic quantum computing”, (2025) arXiv:2411.08955
- [15]
- J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
Page edit log
- Victor V. Albert (2026-05-18) — most recent
Cite as:
“Self-dual CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/self_dual_css