\([[4,1,2]]\) twist-defect code

\([[4,1,2]]\) twist-defect code[1]

Description

A four-qubit non-CSS stabilizer code that can be interpreted as the smallest triangular color code with \(x\)-, \(y\)-, and \(z\)-type Pauli boundaries [2; Fig. 7], and equivalently as a small twist-defect surface code on a tetrahedron inscribed in a sphere [3]. It is the only non-CSS qubit stabilizer code with parameters \([[4,1,2]]\). The code admits weight-three stabilizer generators and weight-two logical Pauli \(X,Y,Z\) operators.

A stabilizer tableau for the code is given by [4; ID 8] \begin{align} \begin{array}{cccc} X & I & X & X \\ Y & Y & I & Y \\ Z & Z & Z & I \end{array}~. \tag*{(1)}\end{align}

Stabilizer generators are shown in Fig. I.

Figure I: Stabilizer generators of the \([[4,1,2]]\) twist-defect code, interpreted as a twist-defect color code. Each boundary type is shown in a different color (red, green, blue), corresponding to an \(X\)-type, \(Y\)-type, or \(Z\)-type Pauli supported on the vertices.

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 [3].

Realizations

Logical Clifford gates were realized in a trapped-ion device by Quantinuum [3].

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 [3].
\([[4,1,2]]\) 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.

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum

Twist-defect color codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum

Parents

Twist-defect color code

A small 6.6.6 twist-defect color code admits three weight-three stabilizer generators [2; Fig. 7]; this code is equivalent to a twist-defect surface code on a tetrahedron inscribed in a sphere [3] via a single-qubit Clifford circuit.

Twist-defect surface codeQLDPC Lattice stabilizer Stabilizer Hamiltonian-based Qubit QECC Quantum

A small 6.6.6 twist-defect color code admits three weight-three stabilizer generators [2; Fig. 7]; this code is equivalent to a twist-defect surface code on a tetrahedron inscribed in a sphere [3] 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]: 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
[3]: S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
[4]: Qiskit Community. Qiskit QEC framework. https://github.com/qiskit-community/qiskit-qec

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/4/stab_4_1_2/stab_4_1_2.yml.