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Efficient Quantum Circuit for Encoding and Decoding of the [[8,3,5]] Stabilizer Code

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Abstract

A general protocol for constructing a complete efficient encoding and decoding quantum circuit of the [[8,3,5]] stabilizer code is proposed. The [[8,3,5]] stabilizer code is an eight-qubit code that protects a three-qubit state with up to one error, which is very important for quantum information processing. Single-qubit operations, two-qubit controlled gates and Toffoli gates are required in the proposed circuit. The current protocol can be generalized to all quantum stabilizer codes satisfying quantum Hamming bound, and implemented in some quantum systems.

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Acknowledgements

This work is supported by National Natural Science Foundation of China (NSFC) under Grant Nos:11005029, 10905024 and 61073048, China Postdoctoral Science Foundation under Grant No. 2011M501373, the Natural Science Foundation of Anhui Province under Grant No. 10040606Q51, the Key Program of the Education Department of Anhui Province under Grant Nos. KJ2011A242, KJ2010ZD08, the Talent Foundation of Anhui Province under Grant No. 2009Z022, the Key Program of Research Base of Hefei Normal University under Grant No. 2012jd13.

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Authors and Affiliations

  1. The Key Laboratory of Intelligent Computing and Signal Processing, Anhui University, Hefei, 230039, China

    Ping Dong & Jun Liu

  2. School of Electronic and Information Engineering, Hefei Normal University, Hefei, 230601, China

    Ping Dong, Jun Liu & Zhuo-Liang Cao

Authors
  1. Ping Dong
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  2. Jun Liu
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  3. Zhuo-Liang Cao
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Corresponding author

Correspondence to Ping Dong.

Appendix: Construction of the Code Words of the [[8, 3, 5]] Stabilizer Code

Appendix: Construction of the Code Words of the [[8, 3, 5]] Stabilizer Code

According to Eq. (3), M 5 has no effect on the generation of the quantum code word of [[8,3,5]] stabilizer code, thus the term (1+M 5) can be eliminated, the code word of the [[8,3,5]] stabilizer code can be reduced as

(7)

Then, we combine Eqs. (6) with (3), the detailed code words of the [[8,3,5]] can be listed as following

According to the above equations, the encoding rule of the [[8,3,5]] stabilizer code can be reduced to

(8)

where m=c 2⊕c 3, \(\overline{m}=n=\overline{c_{2}\oplus c_{3}}\), \(a=(-1)^{c_{1}}\) and \(b=(-1)^{\overline{c}_{1}}\).

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Dong, P., Liu, J. & Cao, ZL. Efficient Quantum Circuit for Encoding and Decoding of the [[8,3,5]] Stabilizer Code. Int J Theor Phys 52, 1274–1281 (2013). https://doi.org/10.1007/s10773-012-1442-x

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