Description
A qubit CSS code whose \(X\)- and \(Z\)-type stabilizer groups are exchanged by a permutation of the physical qubits. Equivalently, a qubit CSS code for which transversal Hadamard followed by a qubit permutation is a logical operation.
Let \(C_X\) and \(C_Z\) be the \(X\)- and \(Z\)-type stabilizer spaces of an \([[n,k]]\) CSS code. The code is permutationally self-dual (PSD) iff there is a permutation \(\pi \in S_n\) with \(\pi(C_X)=C_Z\) and \(\pi(C_Z)=C_X\), equivalently iff the physical operator \(U_\pi H^{\otimes n}\) preserves the stabilizer group [1]. Such a \(\pi\) is called an \(XZ\)-duality [2], and the code is also called em-symmetric [3].
Transversal and Permutation-Based Gates
An \(XZ\)-duality yields Hadamard-type and phase-type fold-transversal Clifford gates, each realized by one layer of physical Clifford gates together with one qubit permutation [2].A qubit permutation composed with transversal Hadamard is a logical Clifford gate. In any CSS logical basis, every such self-duality automorphism induces a pure exchange-type logical Clifford, i.e. one whose symplectic representation has vanishing diagonal blocks [1; Prop. 1].Permutational self-duality is necessary for exchange-type logical actions. An indecomposable CSS code can host logical \(H^{\otimes k}\), \((SH)^{\otimes k}\), \((HS)^{\otimes k}\), or any mixture of these up to composition with logical \(S\), CX and CZ circuits, implemented by single-qubit Cliffords and qubit permutations in a CSS logical basis, only if the code is PSD [1].For an indecomposable PSD-but-not-self-dual code, the logical group generated by single-qubit Cliffords and qubit permutations is \(\mathrm{Trans} \rtimes \mathrm{Perm}.2\) [1].For an indecomposable PSD-but-not-self-dual code, transversal single-qubit Cliffords realize at most \(\mathcal{U}(2\lfloor k/2 \rfloor,2)^2\), acting on two disjoint halves of the logical qubits, so their number is at most \(2^{\lfloor k/2 \rfloor(\lfloor k/2 \rfloor+1)}\) [1].Qubit permutations of a PSD code realize a logical group that preserves a flag of subspaces of \(\mathbb{F}_2^k\). If the all-ones vector lies in \(C_X+C_Z\), then \(n\), \(k\), and \(k-2m\) are even, where \(m\) measures how far the two stabilizer spaces are from coinciding [1].Cousin
- Dual linear code— The \(X\)- and \(Z\)-type stabilizer spaces of a permutationally self-dual CSS code are permutation-equivalent binary linear codes.
Primary Hierarchy
References
- [1]
- J. M. Koh, S. Majidy, A. Chakraborty, A. Gong, S. J. S. Tan, and N. Y. Yao, “Achieving the limits of automorphism gates”, (2026) arXiv:2609.19250
- [2]
- N. P. Breuckmann and S. Burton, “Fold-Transversal Clifford Gates for Quantum Codes”, Quantum 8, 1372 (2024) arXiv:2202.06647 DOI
- [3]
- L. H. English, H. Luo, Y. Wang, B. Srivastava, S. D. Bartlett, and D. J. Williamson, “Duality constrains optimal thresholds in quantum error correction”, (2026) arXiv:2607.21160
Page edit log
- Victor V. Albert (2026-09-19) — most recent
Cite as:
“Permutationally self-dual (PSD) CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/perm_self_dual_css, arXiv:2606.11484