Trapping Sets of Quantum LDPC Codes

Nithin Raveendran and Bane Vasić

Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721, USA

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Abstract

Iterative decoders for finite length quantum low-density parity-check (QLDPC) codes are attractive because their hardware complexity scales only linearly with the number of physical qubits. However, they are impacted by short cycles, detrimental graphical configurations known as trapping sets (TSs) present in a code graph as well as symmetric degeneracy of errors. These factors significantly degrade the decoder decoding probability performance and cause so-called error floor. In this paper, we establish a systematic methodology by which one can identify and classify quantum trapping sets (QTSs) according to their topological structure and decoder used. The conventional definition of a TS from classical error correction is generalized to address the syndrome decoding scenario for QLDPC codes. We show that the knowledge of QTSs can be used to design better QLDPC codes and decoders. Frame error rate improvements of two orders of magnitude in the error floor regime are demonstrated for some practical finite-length QLDPC codes without requiring any post-processing.

Quantum low-density parity-check (QLDPC) codes have recently gained popularity as an important class of quantum error correction codes due to their ability to realize scalable fault-tolerant quantum computers with constant overhead and are decodable using efficient iterative decoders. However, QLDPC code’s decoding performance is impacted by short cycles and detrimental graphical configurations present in their code graph. Such a performance degradation at low noise values – referred to as error floor effect will be severe especially in the case of practically useful finite length QLDPC codes. In classical LDPC coding literature, these harmful configurations classified as $\textit{trapping sets}$ (TSs) have been well studied and have aided to develop low-complexity iterative decoders that surpass the conventional belief propagation decoder. However, TSs have never been formally studied in the context of QLDPC codes and their decoding. In this work, we introduce the concept of $\textit{Quantum Trapping Sets}$ (QTSs) by investigating the failure configurations for syndrome-based iterative decoders. We establish a systematic methodology by which one can identify and classify QTSs according to their topological structure and decoder used. The conventional definition of a TS from classical error correction is generalized to address the syndrome decoding scenario for QLDPC codes. As a summary, we observe two types of QTSs – one is similar to classical TSs and the other is referred to as symmetric stabilizer TSs – these are unique to QLDPC codes. The properties of symmetric stabilizer TSs are distinct and specific to the QLDPC decoding problem and hence, will be instrumental in exploiting the degeneracy of QLDPC codes to the decoder’s advantage. Furthermore, we demonstrate the two advantages of studying QTSs – (1) Design better QLDPC codes – ability to construct QLDPC codes devoid of harmful QTSs, (2) Design better decoders without post-processing steps – ability to devise new decoding algorithms that escape from harmful QTSs and have low error floors.

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► References

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[2] Kao-Yueh Kuo and Ching-Yi Lai, "Exploiting Degeneracy in Belief Propagation Decoding of Quantum Codes", arXiv:2104.13659.

[3] Ching-Yi Lai and Kao-Yueh Kuo, "Log-domain decoding of quantum LDPC codes over binary finite fields", arXiv:2104.00304.

[4] Patricio Fuentes, Josu Etxezarreta Martinez, Pedro M. Crespo, and Javier Garcia-Frias, "On the logical error rate of sparse quantum codes", arXiv:2108.10645.

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