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Fault-tolerant error correction for quantum Hamming codes with only two ancillary qudits

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Abstract

High-dimensional quantum error correction codes (QECCs) are widely used to enhance the security and coding capabilities of the entire quantum systems. While these codes show promise in achieving reliable quantum computing by executing a set of universal quantum gates, quantum gate faults present a challenge for implementing universal quantum computing. The ability to perform universal quantum gates in a fault-tolerant manner is required. In this paper, we propose a fault-tolerant error correction scheme which allows the entire correction process to proceed smoothly, even in the presence of quantum gate faults. Firstly, by equipping the syndrome-extracted circuit with a flag qudit, it is possible to parallel the error detection of coding blocks with the fault detection of quantum gates to detect noise and correlated errors. Then, by giving a feasible permutation of quantum gates, the syndromes can identify the error type and location. We interpret the permutation with an example of Hamming code \([[13,7,3]]_{3}\), a three-dimensional QECC, using only two ancillary qudits. Finally, the results show that our scheme is valid for any quantum Hamming code. The overhead of qudits required in our scheme is independent of the weight and quantity of stabilizer generators, and only two ancillary qudits are involved. Consequently, our scheme is very promising for achieving a low logical error rate in a fault-tolerant manner.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

This work was supported by the NSFC (Grant No. 62271070) and the Fundamental Research Funds for Beijing Municipal Commission of Education.

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Authors and Affiliations

  1. Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, No.10 Xitucheng Road, Beijing, 100876, China

    Li-Yun Zhao, Xiu-Bo Chen, Jing-Wen Zhang & Yi-Xian Yang

  2. School of Information Science and Technology, North China University of Technology, No.5 Jinyuanzhuang Road, Beijing, 100144, China

    Gang Xu

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  1. Li-Yun Zhao
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  2. Xiu-Bo Chen
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  3. Gang Xu
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Correspondence to Xiu-Bo Chen.

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Zhao, LY., Chen, XB., Xu, G. et al. Fault-tolerant error correction for quantum Hamming codes with only two ancillary qudits. Quantum Inf Process 22, 70 (2023). https://doi.org/10.1007/s11128-022-03796-z

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