Fast quantum codes based on Pauli block jacket matrices
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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.
KeywordsPauli block matrix Block jacket transform Pauli matrices Abelian group Quantum error-correction codes
The authors are grateful to the anonymous referees for their detailed suggestions. This work was supported by Natural Science Foundation of Hunan Province (Nos. 07JJ3128, 2008RS4016), Postdoctoral Science Foundation of China (Nos. 20070420184, 200801341), and in part by Joint Project KOSEF/NSFC Korea Research Foundation KRF-2007-521-D00330, Chonbuk National University, Korea.
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- 3.Lee M.H.: The center weighted Hardamard transform. IEEE Trans. Circuits Syst. CAS-36, 1247–1249 (1989)Google Scholar
- 4.Rao K.Y., Hershey J.E.: Hadamard Matrix Analysis and Synthesis. Kluwer, Norwell (1997)Google Scholar
- 8.MacWilliams F.J., Sloane N.J.A.: The Theory of Error Correcting Codes. Elsevier, Amsterdam (1988)Google Scholar
- 10.Song, W., Lee, M.H.: Orthogonal space-time block codes design using jacket transform for MIMO transmission system. In: IEEE International Conference on Communications (ICC 2008), Beijing, China (2008)Google Scholar
- 12.Lee M.H., Finlayson K.: A Simple Element Inverse Jacket Transform Coding. IEEE ITW, New Zealand (2005)Google Scholar
- 15.Nielsen M.A., Chuang I.L.: Quantum computation and quantum information. Cambrige University Press, Cambridge (2002)Google Scholar