中国物理B ›› 2025, Vol. 34 ›› Issue (5): 50311-050311.doi: 10.1088/1674-1056/adcb26

所属专题： SPECIAL TOPIC — Computational programs in complex systems

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Planar: A software for exact decoding quantum error correction codes with planar structure

Dongyang Feng(冯东阳)1,2,3,†, Hanyan Cao(曹涵彦)2,3,†, and Pan Zhang(张潘)1,2,‡   

  1. 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
  • 收稿日期:2025-03-14 修回日期:2025-04-09 接受日期:2025-04-10 出版日期:2025-05-15 发布日期:2025-05-06
  • 通讯作者: Pan Zhang E-mail:panzhang@itp.ac.cn
  • 基金资助:
    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).

Planar: A software for exact decoding quantum error correction codes with planar structure

Dongyang Feng(冯东阳)1,2,3,†, Hanyan Cao(曹涵彦)2,3,†, and Pan Zhang(张潘)1,2,‡   

  1. 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:2025-03-14 Revised:2025-04-09 Accepted:2025-04-10 Online:2025-05-15 Published:2025-05-06
  • Contact: Pan Zhang E-mail:panzhang@itp.ac.cn
  • Supported by:
    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).

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摘要： 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.

关键词: quantum computing, quantum error correction, planar Ising model

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.

Key words: quantum computing, quantum error correction, planar Ising model

中图分类号:  (Quantum error correction and other methods for protection against decoherence)

  • 03.67.Pp
02.50.-r (Probability theory, stochastic processes, and statistics) 05.10.-a (Computational methods in statistical physics and nonlinear dynamics) 05.50.+q (Lattice theory and statistics)