Family of fast quantum stabilizer codes constructions

Markus Grassl

Avanti Ulhas Ketkar

Stabilizer codes form a special class of quantum error correcting codes. Nonbinary quantum stabilizer codes are studied in this thesis. A lot of work on binary quantum stabilizer codes has been done. Nonbinary stabilizer codes have received much less attention. Various results on binary stabilizer codes such as various code families and general code constructions are generalized to the nonbinary case in this thesis. The lower bound on the minimum distance of a code is nothing but the minimum distance of the currently best known code. The focus of this research is to improve the lower bounds on this minimum distance. To achieve this goal, various existing quantum codes are studied that have good minimum distance. Some new families of nonbinary stabilizer codes such as quantum BCH codes are constructed. Different ways of constructing new codes from the existing ones are also found. All these constructions together help improve the lower bounds.

Sujuan Huang, Zhonghua Sun, Shixin Zhu

Rui Chao, Ben W. Reichardt

Conventional fault-tolerant quantum error-correction schemes require a number of extra qubits that grow linearly with the code's maximum stabilizer generator weight.For some common distance-three codes, the recent "flag paradigm" uses just two extra qubits.Chamberland and Beverland [Quantum 2, 53 (2018)] provide a framework for flag error correction of arbitrary-distance codes.However, their construction requires conditions that only some code families are known to satisfy.We give a flag error-correction scheme that works for any stabilizer code, unconditionally.With fast qubit measurement and reset, it uses ≤d + 1 extra qubits for a distance-d code.

Zhuo Li, Zhenhui Zhang, Lijuan Xing

Yuanli Cai, Ying Guo

Motivated by a family of binary cocyclic block matrices over GF(2), we proposed a construction method to gain the stabilizer of long-length quantum error-correction codes (QECCs). Stabilizer quantum codes (SQCs) can be obtained by the different rows of the yielded circulant permutation matrices; hence, the quantum codes have the virtue of a fast construction algorithm. The recursive relation of a block matrix is employed in the proposed approach, so that the generator matrix of quantum cocyclic codes with long length can be constructed easily. Furthermore, the obtained quantum codes have the low-density advantage of there being no 4-cycles in the Tanner graph.