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International Conference on Natural Computation (ICNC)

Multilevel Quantum Error Correction Codes in Transform Domain

Ying Guo, Dazu Huang, Guihua Zeng, Moon Ho Lee

Published Aug 1, 2007• 0 citations
Publisher

Abstract

Quantum error correction codes (QECC) have become cornerstone of quantum information and computation. In this correspondence, we investigate a class of QECC in transform domain. To design the QECC denoted by [[N, k, d]] with larger N, generators of stabilizer of the constructed codes are composed via selecting randomly N - k rows from an yielded matrices with recursive relations of matrices transform. Subsequently, a new kind of stabilizer quantum codes may be generated expediently. In the proposed approach, the parameter N may be arbitrary larger, and the QECC with larger N may be easily constructed due to the employment of the recursive relations of matrices transform. Especially, some new codes, which are impossible to be obtained via the previous approaches, are constructed. The resulting codes are more efficient with better perimeters than the previous quantum codes.

DOI
10.1109/icnc.2007.489

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Quantum error correction codes (QECC) have become cornerstone of quantum information and computation. In this correspondence, we investigate a class of QECC in transform domain. To design the QECC denoted by [[N, k, d]] with larger N, generators of stabilizer of the constructed codes are composed via selecting randomly N - k rows from an yielded matrices with recursive relations of matrices transform. Subsequently, a new kind of stabilizer quantum codes may be generated expediently. In the proposed approach, the parameter N may be arbitrary larger, and the QECC with larger N may be easily constructed due to the employment of the recursive relations of matrices transform. Especially, some new codes, which are impossible to be obtained via the previous approaches, are constructed. The resulting codes are more efficient with better perimeters than the previous quantum codes.
The key problem for constructing a stabilizer quantum code is how to create a set of generators for the stabilizer of the stabilizer quantum code, i.e. check matrix. In this paper, we suggest an approach based on the Clifford algebra to create the check matrix for the stabilizer quantum codes. In the proposed approach, the recursive relation of the matrix transform over the Clifford algebra is employed to generate the check matrix. With the proposed approach, a quantum code with any length can be constructed easily. Especially some new codes, which are impossible via previous approaches, are constructed.
Motivated by the fast block Jacket transform based on Pauli matrices,this paper showed how block Jacket matrix could simplify the coding theory of quantum error-correction.Quantum Jacket codes(QJCs) did not require the dual-containing constraint necessary for standard quantum error-correction codes(QECCs),thus allowing to construct quantum codes of large block length with efficiency.These QJCs were generated from some Abelian subgroups of Pauli group that were composed of different rows of the yielded block Jacket matrix,and had advantages of being fast constructed with asymptotically good behaviors.
It is well known that n-length stabilizer quantum error correcting codes (QECCs) can be obtained via n-length classical error correction codes (CECCs) over GF(4), that are additive and self-orthogonal with respect to the trace Hermitian inner product. But, most of the CECCs have been studied with respect to the Euclidean inner product. In this paper, it is shown that n-length stabilizer QECCs can be constructed via 3n-length linear CECCs over GF(2) that are self-orthogonal with respect to the Euclidean inner product. This facilitates usage of the widely studied self-orthogonal CECCs to construct stabilizer QECCs. Moreover, classical, binary, self-orthogonal cyclic codes have been used to obtain stabilizer QECCs with guaranteed quantum error correcting capability. This is facilitated by the fact that (i) self-orthogonal, binary cyclic codes are easily identified using transform approach and (ii) for such codes lower bounds on the minimum Hamming distance are known. Several explicit codes are constructed including two pure MDS QECCs.
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.
An effective construction method for long-length quantum code has important applications in the field based on large-scale data. With the rapid development of quantum computing, how to construct this class of quantum coding has become one of the key research fields in quantum information theory. Motivated by the block jacket matrix and its circulant permutation, we proposed a construction method for quantum quasi-cyclic (QC) codes with two classical codes. This simplifies the coding process for long-length quantum error-correction code (QECC) using number decomposition. The obtained code length N can achieve O(n2) if an appropriate prime number n is taken. Furthermore, with a suitable parameter in the construction method, the obtained codes have four cycles in their generator matrices and show good performance for low density codes.