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On the hardnesses of several quantum decoding problems

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Abstract

We classify the time complexities of three decoding problems for quantum stabilizer codes: quantum bounded distance decoding (QBDD), quantum maximum likelihood decoding (QMLD), and quantum minimum error probability decoding (QMEPD). For QBDD, we show that it is NP-hard based on Fujita’s result, and cover the gap of full row rank of check matrices, like what Berlekamp, McEliece, and Tilborg suggested in 1978. Then, we give some insight into the quantum decoding problems to clarify that the degeneracy property is implicitly embedded in any decoding algorithm, independent of the typical definition of degenerate codes. Then, over the depolarizing channel model, we show that QMLD and QMEPD are NP-hard. The NP-hardnesses of these decoding problems indicate that decoding general stabilizer codes is extremely difficult, strengthening the foundation of quantum code-based cryptography.

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Funding

This work was supported by the Ministry of Science and Technology, Taiwan, under Contract MOST 101-2221-E-007-096-MY3.

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  1. Department of Electrical Engineering, National Tsing Hua University, Hsinchu, 30013, Taiwan

    Kao-Yueh Kuo & Chung-Chin Lu

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  1. Kao-Yueh Kuo
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  2. Chung-Chin Lu
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Correspondence to Kao-Yueh Kuo.

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This work was supported by the Ministry of Science and Technology, Taiwan, under Contract MOST 101-2221-E-007-096-MY3. Part of this work was presented at the 2012 International Symposium on Information Theory and its Applications (ISITA 2012), Hawaii, USA, October 28–31, 2012. This work has an arXiv version (arxiv:1306.5173).

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Kuo, KY., Lu, CC. On the hardnesses of several quantum decoding problems. Quantum Inf Process 19, 123 (2020). https://doi.org/10.1007/s11128-020-02622-8

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  • DOI: https://doi.org/10.1007/s11128-020-02622-8

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