Abstract
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.
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Acknowledgments
This work has been supported by the National Natural Science Foundation of China (Grant No. 60902044), PhD Program Foundation of Ministry of Education of China (Grant No. 385 20090162120070), Postdoctoral Science Foundation of China (Grant No. 200801341), Science Foundation of Hunan Province (Grant No. 07JJ3128, 2008RS4016), and National Laboratory on Local Fiber-Optical Communication Network and Advanced Optical Communication System of Shanghai Jiaotong University, China.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Shi, R., Guo, Y. & Lee, M.H. Quantum codes based on fast pauli block transforms in the finite field. Quantum Inf Process 9, 611–628 (2010). https://doi.org/10.1007/s11128-010-0171-4
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DOI: https://doi.org/10.1007/s11128-010-0171-4