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International Conference on Natural Computation (ICNC)

Constructions of Quantum Codes Based on Quadratic Residues

Ying Guo, Yangye Liu, Zhigang Chen, Chengrong Huang

Published Jan 1, 2009• 0 citations
Publisher

Abstract

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.

DOI
10.1109/icnc.2009.113

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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.
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.
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.
We propose two types, namely Type-I and Type-II, quantum stabilizer codes using quadratic residue sets of prime modulus given by the form p=4n±1p=4n\pm1p=4n±1. The proposed Type-I stabilizer codes are of cyclic structure and code length N=pN=pN=p. They are constructed based on multi-weight circulant matrix generated from idempotent polynomial, which is obtained from a quadratic residue set. The proposed Type-II stabilizer codes are of quasi-cyclic (QC) structure and code length N=pkN=pkN=pk, where kkk is the size of a quadratic residue set. They are constructed based on structured sparse-graphs codes derived from proto-matrix and circulant permutation matrix. With the proposed methods, we design rich classes of cyclic and quasi-cyclic quantum stabilizer codes with variable code length. We show how the commutative constraint (also referred to as the Symplectic Inner Product constraint) for quantum codes can be satisfied for each proposed construction method. We also analyze both the dimension and distance for Type-I stabilizer codes and the dimension of Type-II stabilizer codes. For the cyclic quantum stabilizer codes, we show that they meet the existing distance bounds in literature.
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.