\([[4,1,2]]\) twist-defect code[1]
Description
A four-qubit non-CSS stabilizer code that can be interpreted as a small twist-defect surface code on a tetrahedron inscribed in a sphere [2]. It is the only non-CSS qubit stabilizer code with parameters \([[4,1,2]]\) [3; ID 8]. The code admits weight-three stabilizer generators \(\{ZIXZ,YXYI,IZZX\}\) and weight-two logical Pauli \(X,Y,Z\) operators.Protection
Detects a single-qubit error or single erasure as a distance-two code.Transversal Gates
Weight-two transversal logical Pauli \(X,Y,Z\) operations [1].Gates
A set of local Clifford operations and permutations (in the twist-defect realization, braiding the four genons) generates the full single-qubit Clifford group [2].Realizations
Logical Clifford gates were realized in a trapped-ion device by Quantinuum [2].Cousins
- \([[4,2,2]]\) Four-qubit code— Adding \(XYZI\) to the stabilizer group of the \([[4,2,2]]\) four-qubit code yields the \([[4,1,2]]\) twist-defect subcode [1].
- Toric code— The symplectic double of the \([[4,1,2]]\) twist-defect code is the \([[8,2,2]]\) twisted toric code [2].
- \([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto (LNCY) code— Adding \(XXII\) (\(XYZI\)) to the stabilizer group of the \([[4,2,2]]\) code yields the \([[4,1,2]]\) LNCY (twist-defect) code.
Primary Hierarchy
Parents
A small 6.6.6 twist-defect color code admits three weight-three stabilizer generators [4; Fig. 7]; this code is equivalent to a twist-defect surface code on a tetrahedron inscribed in a sphere [2] via a single-qubit Clifford circuit.
A small 6.6.6 twist-defect color code admits three weight-three stabilizer generators [4; Fig. 7]; this code is equivalent to a twist-defect surface code on a tetrahedron inscribed in a sphere [2] via a single-qubit Clifford circuit.
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[4,1,2]]\) twist-defect code
References
- [1]
- S. P. Jordan, E. Farhi, and P. W. Shor, “Error-correcting codes for adiabatic quantum computation”, Physical Review A 74, (2006) arXiv:quant-ph/0512170 DOI
- [2]
- S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
- [3]
- A. Cross and D. Vandeth, “Small Binary Stabilizer Subsystem Codes”, (2025) arXiv:2501.17447
- [4]
- M. S. Kesselring, F. Pastawski, J. Eisert, and B. J. Brown, “The boundaries and twist defects of the color code and their applications to topological quantum computation”, Quantum 2, 101 (2018) arXiv:1806.02820 DOI
Page edit log
- Victor V. Albert (2026-04-23) — most recent
Cite as:
“\([[4,1,2]]\) twist-defect code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_4_1_2