Quasi-cyclic quantum code

Quasi-cyclic quantum code[1]

A block code on \(n\) subsystems such that cyclic shifts of the subsystems by \(\ell\geq 1\) leave the codespace invariant.

Cyclic quantum code
Lattice stabilizer code — Lattice stabilizer codes are invariant under translations by a lattice unit cell.
Generalized bicycle (GB) code — An index-\(m\) quasi-cyclic (QC) code of length \(n=m\ell\) is usually defined as a linear code invariant under the \(m\)-step shift permutation \(T_{n}^{m}\).

Quasi-cyclic code
Two-block group-algebra (2BGA) codes — Any Abelian 2BGA code can be thought of as a multi-dimensional index-two quasi-cyclic code. More precisely, any finite Abelian group can be written as a direct product of several cyclic groups, e.g., \(G=C_{m_1}\times C_{m_2}\times \ldots C_{m_D}\) for a product of \(D\) cyclic groups, which is equivalent to a representation \begin{align} G=\langle x_1,\ldots,x_D|x_j^{m_j}=1, x_jx_ix_j^{-1}x_i^{-1}=1, \forall 1\le i,j\le D\rangle. \tag*{(1)}\end{align} Respectively, an element of the group algebra \(\mathbb{F}_q[G]\), where \(\mathbb{F}_q\) is a finite field, can be written as a \(D\)-variate polynomial in \(\mathbb{F}_q[x_1,x_2,\ldots,x_D]\), with the degree of the generator \(x_j\) of order \(m_j\) not exceeding \(m_j-1\). An equivalent construction in terms of Kronecker products of circulant matrices was introduced in [2]. Related higher-dimensional quasi-cyclic and convolutional quantum codes have been constructed in [3].

[1]: M. Hagiwara and H. Imai, “Quantum Quasi-Cyclic LDPC Codes”, 2007 IEEE International Symposium on Information Theory (2007) arXiv:quant-ph/0701020 DOI
[2]: A. A. Kovalev and L. P. Pryadko, “Quantum Kronecker sum-product low-density parity-check codes with finite rate”, Physical Review A 88, (2013) arXiv:1212.6703 DOI
[3]: S. Yang and R. Calderbank, “Spatially-Coupled QDLPC Codes”, (2023) arXiv:2305.00137

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“Quasi-cyclic quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_quasi_cyclic