拓扑量子编码和量子存储：从错误链和纠错阈值到自纠错与低密度量子码

徐桐; 赵宇涛; 刘东

doi:10.7693/wl20260804

物理

拓扑量子编码和量子存储：从错误链和纠错阈值到自纠错与低密度量子码

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徐桐, 赵宇涛, 刘东. 2026: 拓扑量子编码和量子存储：从错误链和纠错阈值到自纠错与低密度量子码, 物理, 55(8): 543-554. doi: 10.7693/wl20260804

引用本文:

徐桐, 赵宇涛, 刘东. 2026: 拓扑量子编码和量子存储：从错误链和纠错阈值到自纠错与低密度量子码, 物理, 55(8): 543-554. doi: 10.7693/wl20260804

XU Tong, ZHAO Yu-Tao, LIU Dong. 2026: Topological quantum codes and memories：from error chains and thresholds to self-correction and quantum low-density parity-check codes, Physics, 55(8): 543-554. doi: 10.7693/wl20260804

Citation:

拓扑量子编码和量子存储：从错误链和纠错阈值到自纠错与低密度量子码

清华大学物理系北京 100084

通讯作者: 刘东,email:dongeliu@mail.tsinghua.edu.cn

Topological quantum codes and memories：from error chains and thresholds to self-correction and quantum low-density parity-check codes

Department of Physics, Tsinghua University, Beijing 100084, China

Corresponding author: LIU Dong, dongeliu@mail.tsinghua.edu.cn

摘要: 量子信息不能任意复制，同时还会受到比特翻转、相位翻转及测量误差的扰动，因此保存未知量子态比保存经典信息更困难。拓扑量子纠错把逻辑信息分散编码在许多物理量子比特中，使局域错误留下可测的错误症候；解码器再依据这些症候判断恢复操作是否会形成跨越系统的逻辑错误。纠错阈值说明何时扩大码距能持续降低逻辑错误率，也给出了可扩展量子存储的基本条件。文章从稳定子、错误链与同调的直观图像出发，介绍表面码的主动纠错和近期实验，并讨论低维局域性为何限制编码开销，进而引出被动自纠错与良好量子低密度奇偶校验码两条互补路线。
- 拓扑量子纠错 /
- 表面码 /
- 纠错阈值 /
- 量子存储 /
- 自纠错 /
- 量子低密度奇偶校验码
Abstract: Quantum information cannot be copied arbitrarily and is vulnerable to bitflip, phase-flip, and measurement errors, making the preservation of an unknown quantum state fundamentally more demanding than the storage of classical information. Topological quantum error correction protects logical information by distributing it nonlocally across many physical qubits, so that local faults produce measurable error syndromes. A decoder uses these syndromes to infer an appropriate recovery operation while avoiding system-spanning error chains that implement nontrivial logical operators. The error-correction threshold identifies the regime in which increasing the code distance leads to a sustained reduction in the logical error rate and therefore provides a basic criterion for scalable quantum memories. Starting from an intuitive description of stabilizers, error chains, and homology, we present a review of active error correction with surface codes and recent experimental progress. We then discuss how low-dimensional locality places fundamental constraints on coding overhead, motivating two complementary directions: passive self-correcting quantum memories and good quantum low-density parity-check codes.
- topological quantum error correction /
- surface codes /
- error-correction threshold /
- quantum memory /
- self-correcting /
- quantum low-density parity-check codes .

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拓扑量子编码和量子存储：从错误链和纠错阈值到自纠错与低密度量子码

通讯作者: 刘东,email:dongeliu@mail.tsinghua.edu.cn

清华大学物理系北京 100084

收稿日期: 2026-07-15

关键词:

摘要: 量子信息不能任意复制，同时还会受到比特翻转、相位翻转及测量误差的扰动，因此保存未知量子态比保存经典信息更困难。拓扑量子纠错把逻辑信息分散编码在许多物理量子比特中，使局域错误留下可测的错误症候；解码器再依据这些症候判断恢复操作是否会形成跨越系统的逻辑错误。纠错阈值说明何时扩大码距能持续降低逻辑错误率，也给出了可扩展量子存储的基本条件。文章从稳定子、错误链与同调的直观图像出发，介绍表面码的主动纠错和近期实验，并讨论低维局域性为何限制编码开销，进而引出被动自纠错与良好量子低密度奇偶校验码两条互补路线。

English Abstract

Topological quantum codes and memories：from error chains and thresholds to self-correction and quantum low-density parity-check codes

Corresponding author: LIU Dong, dongeliu@mail.tsinghua.edu.cn

清华大学物理系北京 100084

Received Date: 2026-07-15

Keywords:

Abstract: Quantum information cannot be copied arbitrarily and is vulnerable to bitflip, phase-flip, and measurement errors, making the preservation of an unknown quantum state fundamentally more demanding than the storage of classical information. Topological quantum error correction protects logical information by distributing it nonlocally across many physical qubits, so that local faults produce measurable error syndromes. A decoder uses these syndromes to infer an appropriate recovery operation while avoiding system-spanning error chains that implement nontrivial logical operators. The error-correction threshold identifies the regime in which increasing the code distance leads to a sustained reduction in the logical error rate and therefore provides a basic criterion for scalable quantum memories. Starting from an intuitive description of stabilizers, error chains, and homology, we present a review of active error correction with surface codes and recent experimental progress. We then discuss how low-dimensional locality places fundamental constraints on coding overhead, motivating two complementary directions: passive self-correcting quantum memories and good quantum low-density parity-check codes.

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拓扑量子编码和量子存储：从错误链和纠错阈值到自纠错与低密度量子码

通讯作者: 刘东,email:dongeliu@mail.tsinghua.edu.cn

Topological quantum codes and memories：from error chains and thresholds to self-correction and quantum low-density parity-check codes

Corresponding author: LIU Dong, dongeliu@mail.tsinghua.edu.cn

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拓扑量子编码和量子存储：从错误链和纠错阈值到自纠错与低密度量子码

通讯作者: 刘东,email:dongeliu@mail.tsinghua.edu.cn

English Abstract

Topological quantum codes and memories：from error chains and thresholds to self-correction and quantum low-density parity-check codes

Corresponding author: LIU Dong, dongeliu@mail.tsinghua.edu.cn

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