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Dongyang Feng(冯东阳), Hanyan Cao(曹涵彦), and Pan Zhang(张潘). 2025: Planar: A software for exact decoding quantum error correction codes with planar structure, Chinese Physics B, 34(5): 050311. doi: 10.1088/1674-1056/adcb26
Citation: Dongyang Feng(冯东阳), Hanyan Cao(曹涵彦), and Pan Zhang(张潘). 2025: Planar: A software for exact decoding quantum error correction codes with planar structure, Chinese Physics B, 34(5): 050311. doi: 10.1088/1674-1056/adcb26
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Planar: A software for exact decoding quantum error correction codes with planar structure

  • 1 School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China;

  • 2 CAS Key Laboratory for Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China;

  • 3 School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

  • Received Date: 14/03/2025
    Accepted Date: 09/04/2025
  • Fund Project:

    This work is supported by the National Natural Science Foundation of China (Grant Nos. 12325501, 12047503, and 12247104) and the Chinese Academy of Sciences (Grant No. ZDRW-XX-2022-3-02). P. Z. is partially supported by the Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0301900).

  • PACS: 03.67.Pp; 02.50.-r; 05.10.-a; 05.50.+q

  • Quantum error correction is essential for realizing fault-tolerant quantum computing, where both the efficiency and accuracy of the decoding algorithms play critical roles. In this work, we introduce the implementation of the Planar algorithm, a software framework designed for fast and exact decoding of quantum codes with a planar structure. The algorithm first converts the optimal decoding of quantum codes into a partition function computation problem of an Ising spin glass model. Then it utilizes the exact Kac-Ward formula to solve it. In this way, Planar offers the exact maximum likelihood decoding in polynomial complexity for quantum codes with a planar structure, including the surface code with independent code-capacity noise and the quantum repetition code with circuit-level noise. Unlike traditional minimum-weight decoders such as minimum-weight perfect matching (MWPM), Planar achieves theoretically optimal performance while maintaining polynomial-time efficiency. In addition, to demonstrate its capabilities, we exemplify the implementation using the rotated surface code, a commonly used quantum error correction code with a planar structure, and show that Planar achieves a threshold of approximately $ p_{\rm uc} \approx 0.109 $ under the depolarizing error model, with a time complexity scaling of $ O(N^{0.69}) $, where $ N $ is the number of spins in the Ising model.
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Planar: A software for exact decoding quantum error correction codes with planar structure

  • Received Date: 14/03/2025
  • Accepted Date: 09/04/2025
Fund Project: 

Abstract: Quantum error correction is essential for realizing fault-tolerant quantum computing, where both the efficiency and accuracy of the decoding algorithms play critical roles. In this work, we introduce the implementation of the Planar algorithm, a software framework designed for fast and exact decoding of quantum codes with a planar structure. The algorithm first converts the optimal decoding of quantum codes into a partition function computation problem of an Ising spin glass model. Then it utilizes the exact Kac-Ward formula to solve it. In this way, Planar offers the exact maximum likelihood decoding in polynomial complexity for quantum codes with a planar structure, including the surface code with independent code-capacity noise and the quantum repetition code with circuit-level noise. Unlike traditional minimum-weight decoders such as minimum-weight perfect matching (MWPM), Planar achieves theoretically optimal performance while maintaining polynomial-time efficiency. In addition, to demonstrate its capabilities, we exemplify the implementation using the rotated surface code, a commonly used quantum error correction code with a planar structure, and show that Planar achieves a threshold of approximately $ p_{\rm uc} \approx 0.109 $ under the depolarizing error model, with a time complexity scaling of $ O(N^{0.69}) $, where $ N $ is the number of spins in the Ising model.

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