We introduce a framework based on classical quasi group codes to define a class of quantum CSS codes, called quantum group codes, supporting transversal multi-control- $Z$ gates which are both addressable and parallelizable, thus allowing to efficiently implement circuits composed of non-Clifford gates at the logical level. Building on this, we use a lifting procedure of classical AG codes established from class field theory to construct good quantum group codes with improved decoding complexity and logical multi-control- $Z$ gate parallelizability. More precisely, on input a good quantum AG code over the alphabet $\mathbb F_q$ with transversal $\mathsf{C}^m\mathsf Z$ gate, we apply this lifting procedure to its underlying classical AG code and obtain a quantum group code over the alphabet $\mathbb F_{q^2}$ supporting a transversal $\mathsf{C}^m\mathsf Z$ gate as well as addressable and parallelizable $\mathsf{C}^{m-1}\mathsf Z$ gates. In addition, this quantum code admits a quasi-quadratic time decoder with a linear decoding radius. This is to be compared with the previous quantum AG codes which have a cubic-time decoder. Hence, our work implies a decrease of the time complexity of state-of-the-art magic-state distillation protocols by an almost linear factor.We introduce a framework based on classical quasi group codes to define a class of quantum CSS codes, called quantum group codes, supporting transversal multi-control-$Z$ gates which are both addressable and parallelizable, thus allowing to efficiently implement circuits composed of non-Clifford gates at the logical level. Building on this, we use a lifting procedure of classical AG codes established from class field theory to construct good quantum group codes with improved decoding complexity and logical multi-control-$Z$ gate parallelizability. More precisely, on input a good quantum AG code over the alphabet $\mathbb F_q$ with transversal $\mathsf{C}^m\mathsf Z$ gate, we apply this lifting procedure to its underlying classical AG code and obtain a quantum group code over the alphabet $\mathbb F_{q^2}$ supporting a transversal $\mathsf{C}^m\mathsf Z$ gate as well as addressable and parallelizable $\mathsf{C}^{m-1}\mathsf Z$ gates. In addition, this quantum code admits a quasi-quadratic time decoder with a linear decoding radius. This is to be compared with the previous quantum AG codes which have a cubic-time decoder. Hence, our work implies a decrease of the time complexity of state-of-the-art magic-state distillation protocols by an almost linear factor.

quant-pharXiv:2606.27211Jun 26, 20260 citations

Quantum Error-Correcting Codes Over Small Fields From AG Codes

Heeralal Janwa, Fernando Piñero González

Springer proceedings in mathematics & statisticsJan 1, 20241 citation

Integrated parallel interleaved concatenation for lowering error floors of LDPC codes

Naoaki Kokubun, Hironori Uchikawa

In order to suppress error floors of low density parity check (LDPC) codes, we study a parallel concatenation scheme with an intermediate single parity check (SPC) code and interleavers. The interleavers of the concatenation play an important role to lower error floors, because it is highly possible that interleaved bits in a trapping set are not to be in error even if the bits in the trapping set are originally in error. For quasi-cyclic LDPC codes, our simulation results confirm that the integrated parallel interleaved concatenation with circulant-size cyclic-shift interleavers improves error-floor performance, and it is better than serial concatenation schemes with BCH codes. If a decoder can afford to accommodate higher-hardware complexity, the decoding performance of our scheme improves so that the error floor can be lowered further.In order to suppress error floors of low density parity check (LDPC) codes, we study a parallel concatenation scheme with an intermediate single parity check (SPC) code and interleavers. The interleavers of the concatenation play an important role to lower error floors, because it is highly possible that interleaved bits in a trapping set are not to be in error even if the bits in the trapping set are originally in error. For quasi-cyclic LDPC codes, our simulation results confirm that the integrated parallel interleaved concatenation with circulant-size cyclic-shift interleavers improves error-floor performance, and it is better than serial concatenation schemes with BCH codes. If a decoder can afford to accommodate higher-hardware complexity, the decoding performance of our scheme improves so that the error floor can be lowered further.

IEEE International Symposium on Information Theory (ISIT)Jul 1, 20161 citation

Decoding Methods for Linear Codes

Kenny Barrese

Jun 4, 20080 citations

Iterative algebraic decoding of codes defined on graphs

Xiangyu Tang

Furthermore, we propose novel error correction coding schemes, called Generalized Integrated Interleaving and Sparsely Integrated Interleaving codes. In the context of block interleaved codewords, Generalized Integrated Interleaving allows nonuniform redundancy to be shared among all the interleaves. This allows the redundancy to be adjusted on-the-fly to better suit the error statistics of the channel or storage device. Sparsely Integrated Interleaving groups data nodes in a distributed storage system into subgroups. A data node can belong to several subgroups. A localized algebraic iterative decoding algorithm is used to decode across subgroups to correct large errors. Very little correction capability is sacrificed to achieve fast error correction and lower communication overhead. This scheme improves data access for all the data nodes and allows easy scaling of the distributed storage network.Furthermore, we propose novel error correction coding schemes, called Generalized Integrated Interleaving and Sparsely Integrated Interleaving codes. In the context of block interleaved codewords, Generalized Integrated Interleaving allows nonuniform redundancy to be shared among all the interleaves. This allows the redundancy to be adjusted on-the-fly to better suit the error statistics of the channel or storage device. Sparsely Integrated Interleaving groups data nodes in a distributed storage system into subgroups. A data node can belong to several subgroups. A localized algebraic iterative decoding algorithm is used to decode across subgroups to correct large errors. Very little correction capability is sacrificed to achieve fast error correction and lower communication overhead. This scheme improves data access for all the data nodes and allows easy scaling of the distributed storage network.

Jan 1, 20080 citations

We introduce a framework based on classical quasi group codes to define a class of quantum CSS codes, called quantum group codes, supporting transversal multi-control-

Z

gates which are both addressable and parallelizable, thus allowing to efficiently implement circuits composed of non-Clifford gates at the logical level. Building on this, we use a lifting procedure of classical AG codes established from class field theory to construct good quantum group codes with improved decoding complexity and logical multi-control-

Z

gate parallelizability. More precisely, on input a good quantum AG code over the alphabet

\mathbb F_q

with transversal

\mathsf{C}^m\mathsf Z

gate, we apply this lifting procedure to its underlying classical AG code and obtain a quantum group code over the alphabet

\mathbb F_{q^2}

supporting a transversal

\mathsf{C}^m\mathsf Z

gate as well as addressable and parallelizable

\mathsf{C}^{m-1}\mathsf Z

gates. In addition, this quantum code admits a quasi-quadratic time decoder with a linear decoding radius. This is to be compared with the previous quantum AG codes which have a cubic-time decoder. Hence, our work implies a decrease of the time complexity of state-of-the-art magic-state distillation protocols by an almost linear factor.