\((5,1,2)\)-convolutional code[1]
Description
Family of quantum convolutional codes that are 1D lattice generalizations of the five-qubit perfect code, with the former''s lattice-translation symmetry being the extension of the latter''s cyclic permutation symmetry.
Their stabilizer generators for semi-open boundary conditions are \begin{align} \begin{array}{cccccccc} X & Z & I & I & I & I & I & \cdots\\ Z & X & X & Z & I & I & I & \cdots\\ I & Z & X & X & Z & I & I & \cdots\\ I & I & Z & X & X & Z & I & \cdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array}~. \tag*{(1)}\end{align}
Parent
Child
- Five-qubit perfect code — The \((5,1,2)\)-convolutional code is 1D lattice extension of the five-qubit perfect code, with the former's lattice-translation symmetry being the extension of the latter's cyclic permutation symmetry. The \((5,1,2)\)-convolutional code code reduces to the five-qubit code for a five-qubit chain and periodic boundary conditions. See Ref. [2] for the first few codes in a different extension of the five-qubit perfect code.
Cousin
- Twisted XZZX toric code — \((5,1,2)\)-convolutional codes (twisted XZZX toric codes) are 1D (2D) lattice extensions of the five-qubit perfect code.
References
- [1]
- H. Ollivier and J.-P. Tillich, “Description of a Quantum Convolutional Code”, Physical Review Letters 91, (2003) arXiv:quant-ph/0304189 DOI
- [2]
- Ilya. A. Simakov and Ilya. S. Besedin, “Low-overhead quantum error correction codes with a cyclic topology”, (2024) arXiv:2211.03094
Page edit log
- Victor V. Albert (2023-01-04) — most recent
Cite as:
“\((5,1,2)\)-convolutional code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/stab_5_1_2_convolutional