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.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.

Nov 1, 20050 citations

Family of fast quantum stabilizer codes constructions

Yuanli Cai, Xingxiang Liu

Optical ReviewMar 1, 20100 citations

New construction of binary and nonbinary quantum stabilizer codes based on symmetric matrices

Duc Manh Nguyen, Sunghwan Kim

In this paper, we propose two construction methods for binary and nonbinary quantum stabilizer codes based on symmetric matrices. In the first construction, we use the identity and symmetric matrices to generate parity-check matrices that satisfy the symplectic inner product (SIP) for the construction of quantum stabilizer codes. In the second construction, we modify the first construction to generate parity-check matrices based on the Calderbank–Shor–Stean structure for the construction of quantum stabilizer codes. The binary and nonbinary quantum stabilizer codes whose parameters achieve equality of the quantum singleton bound are investigated with the code lengths ranging from 4 to 12.In this paper, we propose two construction methods for binary and nonbinary quantum stabilizer codes based on symmetric matrices. In the first construction, we use the identity and symmetric matrices to generate parity-check matrices that satisfy the symplectic inner product (SIP) for the construction of quantum stabilizer codes. In the second construction, we modify the first construction to generate parity-check matrices based on the Calderbank–Shor–Stean structure for the construction of quantum stabilizer codes. The binary and nonbinary quantum stabilizer codes whose parameters achieve equality of the quantum singleton bound are investigated with the code lengths ranging from 4 to 12.

International Journal of Modern Physics BSep 30, 20197 citations

Graphical nonbinary quantum error-correcting codes

Dan Mei Hu, Weidong Tang, Meisheng Zhao +3 more

In this paper, we present a systematic way based on the nonbinary graph state of constructing good nonbinary quantum codes, both additive and nonadditive, for systems with integer dimensions. With a computer search, which results in many interesting codes including some nonadditive codes meeting the Singleton bounds, we are able to construct explicitly four families of optimal codes, namely, ${[[6,2,3]]}_{p}$ , ${[[7,3,3]]}_{p}$ , ${[[8,2,4]]}_{p}$ , and ${[[8,4,3]]}_{p}$ for any odd dimension $\mathit{p}$ and a family of nonadditive codes ${((5,p,3))}_{p}$ for arbitrary $p>3$ . In the case of composite numbers as dimensions, we also construct a family of stabilizer codes ${((6,2{p}^{2},3))}_{2p}$ for odd $\mathit{p}$ , whose coding subspace is not of a dimension that is a power of the dimension of the physical subsystem.In this paper, we present a systematic way based on the nonbinary graph state of constructing good nonbinary quantum codes, both additive and nonadditive, for systems with integer dimensions. With a computer search, which results in many interesting codes including some nonadditive codes meeting the Singleton bounds, we are able to construct explicitly four families of optimal codes, namely, ${[[6,2,3]]}_{p}$, ${[[7,3,3]]}_{p}$, ${[[8,2,4]]}_{p}$, and ${[[8,4,3]]}_{p}$ for any odd dimension $\mathit{p}$ and a family of nonadditive codes ${((5,p,3))}_{p}$ for arbitrary $p>3$. In the case of composite numbers as dimensions, we also construct a family of stabilizer codes ${((6,2{p}^{2},3))}_{2p}$ for odd $\mathit{p}$, whose coding subspace is not of a dimension that is a power of the dimension of the physical subsystem.

arXiv:0801.0831Physical Review AJul 3, 200875 citations

Constructions of New Nonbinary Quantum Codes

Xueqin Hu, Guanghui Zhang, Bocong Chen

International Journal of Theoretical PhysicsJun 6, 201422 citations

{[[6,2,3]]}_{p}

{[[7,3,3]]}_{p}

{[[8,2,4]]}_{p}

, and

{[[8,4,3]]}_{p}

for any odd dimension

\mathit{p}

and a family of nonadditive codes

{((5,p,3))}_{p}

for arbitrary

p>3

. In the case of composite numbers as dimensions, we also construct a family of stabilizer codes

{((6,2{p}^{2},3))}_{2p}

for odd

\mathit{p}

, whose coding subspace is not of a dimension that is a power of the dimension of the physical subsystem.