Alternative Names: Weakly self-dual CSS code, Symmetric CSS code, Self-orthogonal CSS code, Homogeneous CSS code, X/Z symmetric CSS code.
Description
A qubit CSS code for which transversal Hadamard is a logical operation. Equivalently, a qubit CSS code constructed from a Euclidean self-orthogonal code.
Specializing the CSS construction to the case when \(C_Z=[n,k]\) is dual-containing (equivalently, \(C_Z^{\perp}\) is self-orthogonal) and \(C_X=C_Z\) yields an \([[n,2k-n]]\) self-dual qubit CSS code. Its \(X\)-type and \(Z\)-type stabilizers are identically supported, and transversal Hadamard preserves the stabilizer group. Such codes are also called weakly self-dual, symmetric [1], self-orthogonal, or homogeneous [2] qubit CSS codes.
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 and Permutation-Based 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 \(Sp(2\ell,2)\) 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 \(m\), this gate together with transversal Hadamard and CNOT yields the full logical Clifford group transversally [7][8; Lemma I.4]. Transversal single-qubit Clifford gates on an indecomposable self-dual CSS code realize exactly the logical group \(S_3\), independently of \(n\), \(k\), and \(d\) [9].Among indecomposable CSS codes, only self-dual codes with \(k=1\) generate the full logical Clifford group across several blocks using single-qubit Clifford gates, qubit permutations, and transversal interblock CX gates [9; Thm. 7]. Direct sums of such single-logical-qubit codes can also generate the full group [9; Rem. 11].Fault Tolerance
Any self-dual CSS code with bounded-weight stabilizer generators admits flag fault-tolerant syndrome extraction [10].Triorthogonal codes realizing logical \(T\) gates using only physical \(T\) gates can be paired up with self-dual CSS codes. Such a pair yields 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 a.k.a. weakly self-dual) binary linear codes.
- Qubit stabilizer code— Any \([[n,k,d]]\) qubit stabilizer code maps to a \([[4n,2k,2d]]\) self-dual CSS code under concatenated symplectic doubling [11; Corr. 2][12; Corr. 1]. Equivalently, one first concatenates each qubit with the tetron code to obtain an intermediate \([[2n,k,2d]]_{f}\) Majorana stabilizer code, and then assigns one qubit to each Majorana mode [12; Lemma 2].
- Tetron code— Any \([[n,k,d]]\) qubit stabilizer code maps to a \([[4n,2k,2d]]\) self-dual CSS code under concatenated symplectic doubling [11; Corr. 2][12; Corr. 1]. Equivalently, one first concatenates each qubit with the tetron code to obtain an intermediate \([[2n,k,2d]]_{f}\) Majorana stabilizer code, and then assigns one qubit to each Majorana mode [12; Lemma 2].
- \([[4,2,2]]\) Four-qubit code— Any \([[n,k,d]]\) qubit stabilizer code maps to a \([[4n,2k,2d]]\) self-dual CSS code under concatenated symplectic doubling [11; Corr. 2][12; Corr. 1]. Equivalently, one first concatenates each qubit with the tetron code to obtain an intermediate \([[2n,k,2d]]_{f}\) Majorana stabilizer code, and then assigns one qubit to each Majorana mode [12; Lemma 2].
- 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 [13,14].
- Quantum divisible code— 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 \(m\), this gate together with transversal Hadamard and CNOT yields the full logical Clifford group transversally [7][8; Lemma I.4].
- Generalized quantum divisible code— Any self-dual CSS code yields a level-three generalized quantum divisible code when level-lifted [15; 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. Such a pair yields 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. The conversion replaces qubit \(Z\)-type and \(X\)-type operators with \(\gamma\)-type and \(\tilde{\gamma}\)-type Majorana operators, respectively [16].
- \([[8,2,3]]\) Hermitian code— Applying concatenated symplectic doubling to the \([[8,2,3]]\) Hermitian code yields a \([[32,4,6]]\) self-dual CSS code [11; Corr. 2].
- 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 the permutationally self-dual CSS codes for which the \(XZ\)-duality can be taken to be the identity permutation. A PSD code on \(2n\) qubits with a fixed-point-free involutory \(XZ\)-duality becomes self-dual after concatenation with \(n\) copies of the \([[4,2,2]]\) code along the duality [17; Thm. 3.5].
If \(H_X = H_Z = C\), then the associated additive \(\mathbb{F}_4\) stabilizer code is \(C + \omega C \subset \mathbb{F}_4^n\). This set is closed under multiplication by \(\omega\) because \(\omega(a + \omega b) = b + \omega(a+b)\), and \(a,b,a+b \in C\), so the code is \(\mathbb{F}_4\)-linear.
Self-dual CSS code
Children
Quantum Logic Codes are built from self-dual CSS cores and the tiling and Steane-concatenation operations preserve self-duality [18].
The code is a hyperbolic self-dual CSS code: the column weight \(J=5\) is odd, so the all-ones vector lies in each row space and every logical operator has even weight, forcing the pairing matrix to be alternating in every basis and hence admitting no normal magic basis [19][3; Sec. III]. Consistently, transversal Hadamard acts pairwise rather than as thirty logical Hadamards, and \(n\), \(k\), and \(d\) are all even.
Bicycle codes use the same check matrix \(H_X=H_Z=(A|A^T)\) for both check types [20]. Their \(X\)- and \(Z\)-type stabilizer spaces therefore coincide.
The \([[22,2,6]]\) shortened Golay code is the self-dual CSS code of the doubly even self-orthogonal \([22,10,8]\) code obtained by shortening the extended Golay code on two bits [3; Appx. A.1.5].
Each concatenation level is a self-dual CSS code, since transversal physical Hadamard implements the logical Hadamard at every level [5; Corrs. 1 and 2].
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 [21; Sec. 7.15.3].
References
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Page edit log
- Victor V. Albert (2026-09-26) — most recent
- Victor V. Albert (2026-09-20)
- Victor V. Albert (2026-09-06)
- Victor V. Albert (2026-08-25)
- Victor V. Albert (2026-06-08)
- Victor V. Albert (2026-05-18)
Cite as:
“Self-dual CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/self_dual_css, arXiv:2606.11484