Quantum Error-Control Codes

The article surveys quantum error control, focusing on quantum stabilizer codes, stressing on the how to use classical codes to design good quantum codes. It is to appear as a book chapter in "A Concise Encyclopedia of Coding Theory," edited by C. Huffman, P. Sole and J-L Kim, to be published by CRC Press

Locally recoverable codes from the matrix-product construction

Matrix-product constructions giving rise to locally recoverable codes are considered, both the classical $r$ and $(r,\delta)$ localities. We study the recovery advantages offered by the constituent codes and also by the defining matrices of the matrix product codes. Finally, we extend these methods to a particular variation of matrix-product codes and quasi-cyclic codes. Singleton-optimal locally recoverable codes and almost Singleton-optimal codes, with length larger than the finite field size, are obtained, some of them with superlinear length

Symplectic self-orthogonal quasi-cyclic codes

In this paper, we obtain sufficient and necessary conditions for quasi-cyclic codes with index even to be symplectic self-orthogonal. Then, we propose a method for constructing symplectic self-orthogonal quasi-cyclic codes, which allows arbitrary polynomials that coprime $x^{n}-1$ to construct symplectic self-orthogonal codes. Moreover, by decomposing the space of quasi-cyclic codes, we provide lower and upper bounds on the minimum symplectic distances of a class of 1-generator quasi-cyclic codes and their symplectic dual codes. Finally, we construct many binary symplectic self-orthogonal codes with excellent parameters, corresponding to 117 record-breaking quantum codes, improving Grassl's table (Bounds on the Minimum Distance of Quantum Codes. http://www.codetables.de)

Quantum two-block group algebra codes

We consider quantum two-block group algebra (2BGA) codes, a previously unstudied family of smallest lifted-product (LP) codes. These codes are related to generalized-bicycle (GB) codes, except a cyclic group is replaced with an arbitrary finite group, generally non-abelian. As special cases, 2BGA codes include a subset of square-matrix LP codes over abelian groups, including quasi-cyclic codes, and all square-matrix hypergraph-product codes constructed from a pair of classical group codes. We establish criteria for permutation equivalence of 2BGA codes and give bounds for their parameters, both explicit and in relation to other quantum and classical codes. We also enumerate the optimal parameters of all inequivalent connected 2BGA codes with stabilizer generator weights $W \le 8$, of length $n \le 100$ for abelian groups, and $n \le 200$ for non-abelian groups.Comment: 19 pages, 9 figures, 3 table

Distance bounds for generalized bicycle codes

Generalized bicycle (GB) codes is a class of quantum error-correcting codes constructed from a pair of binary circulant matrices. Unlike for other simple quantum code ans\"atze, unrestricted GB codes may have linear distance scaling. In addition, low-density parity-check GB codes have a naturally overcomplete set of low-weight stabilizer generators, which is expected to improve their performance in the presence of syndrome measurement errors. For such GB codes with a given maximum generator weight $w$, we constructed upper distance bounds by mapping them to codes local in $D\le w-1$ dimensions, and lower existence bounds which give $d\ge {\cal O}({n}^{1/2})$. We have also done an exhaustive enumeration of GB codes for certain prime circulant sizes in a family of two-qubit encoding codes with row weights 4, 6, and 8; the observed distance scaling is consistent with $A(w){n}^{1/2}+B(w)$, where $n$ is the code length and $A(w)$ is increasing with $w$.Comment: 12 pages, 5 figure

Quasi-cyclic constructions of quantum codes

We give sufficient conditions for self-orthogonality with respect to symplec- tic, Euclidean and Hermitian inner products of a wide family of quasi-cyclic codes of index two. We provide lower bounds for the symplectic weight and the minimum distance of the involved codes. Supported in the previous results, we show algebraic constructions of good quantum codes and determine their parameter