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International Conference on Natural Computation (ICNC)

Fast Stabilizer Quantum Codes Using Pauli Block Matrices

ChenRrong Huang, Ying Guo, Moon-Ho Lee

Published Jan 1, 2008• 0 citations
Publisher

Abstract

Based on the fast block transform of Pauli matrices, we show how Pauli block matrix can simplify the coding theory of quantum error-correction. These SQCs are generated from some abelian subgroups of Pauli group that are composed of different rows of the yielded Pauli block matrix, and hence have advantages of being fast constructed with asymptotically good behaviors.

DOI
10.1109/icnc.2008.407

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Based on the fast block transform of Pauli matrices, we show how Pauli block matrix can simplify the coding theory of quantum error-correction. These SQCs are generated from some abelian subgroups of Pauli group that are composed of different rows of the yielded Pauli block matrix, and hence have advantages of being fast constructed with asymptotically good behaviors.
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.
Jacket matrices motivated by the center weight Hadamard matrices have played an important role in signal processing, communications, image compression, cryptography, etc. In this paper, we suggest a design approach for the Pauli block jacket matrix achieved by substituting some Pauli matrices for all elements of common matrices. Since, the well-known Pauli matrices have been widely utilized for quantum information processing, the large-order Pauli block jacket matrix that contains commutative row operations are investigated in detail. After that some special Abelian groups are elegantly generated from any independent rows of the yielded Pauli block jacket matrix. Finally, we show how the Pauli block jacket matrix can simplify the coding theory of quantum error-correction. The quantum codes we provide do not require the dual-containing constraint necessary for the standard quantum error-correction codes, thus allowing us to construct quantum codes of the large codeword length. The proposed codes can be constructed structurally by using the stabilizer formalism of Abelian groups whose generators are selected from the row operations of the Pauli block jacket matrix, and hence have advantages of being fast constructed with the asymptotically good behaviors.
Motivated by the fast Pauli block transforms (or matrices) over the finite field GF ( q ) for an arbitrary number q , we suggest how to construct the simplified quantum code on the basis of quadratic residues. The present quantum code, which is the stabilizer quantum code, can be fast generated from an Abelian group with commutative quantum operators being selected from a suitable Pauli block matrix. This construction does not require the dual-containing or self-orthogonal constraint for the standard quantum error-correction code, thus allowing us to construct a quantum code with much efficiency.
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.
In this paper, we illustrate how to simplify the constructions of quantum error-correction codes via the quadratic residues. The suggested quantum code, which is the stabilizer quantum code generated from an Abelian group with the basis being commutative quantum operators selected from rows of an Pauli block matrix, does not require the dual-containing (or self-orthogonal) constraint necessary for the standard quantum error-correction code, thus allowing us to construct a quantum code with the large codeword length.