拓扑量子码正相干错误阈值的证明
Proof of a positive coherent-error threshold for topological quantum codes
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中文总结 AI 辅助
该论文严格证明了拓扑量子码（如表面码）在相干Z旋转错误下存在正阈值，通过结合傅里叶分析和聚合物模型簇展开，展示了最大似然恢复在码距离上指数抑制纠缠不保真度，扩展了量子纠错的理论基础。
中文摘要 AI 辅助
量子纠错码的阈值分析在随机错误模型下已经得到了很好的建立，在随机错误模型中，错误以给定的概率随机发生。然而，实际设备中的错误也可能是相干的，例如由于控制不完善而产生的不想要的$Z$旋转，这些错误在随机错误模型中并未被捕获。对于表面码，数值研究表明即使在相干错误下也存在阈值行为，但缺乏阈值存在的严格证明。在此，我们证明对于具有有界逻辑量子比特数的量子低密度奇偶校验码（包括表面码和其他拓扑码），存在相干$Z$旋转错误的正阈值。具体来说，我们表明，当旋转角度低于一个与码大小无关的常数时，最大似然泡利恢复将纠缠不保真度按码距离指数级抑制，直至物理量子比特数线性前因子。该证明结合了傅里叶分析以保留相干错误振幅间的干涉，以及抽象聚合物模型的簇展开。我们的结果扩展了量子纠错的理论基础，并为超越随机错误的量子纠错提供了统计力学描述。
英文摘要
Threshold analyses of quantum error-correcting codes are well established for stochastic error models, in which errors occur randomly with given probabilities. However, errors in actual devices can also be coherent, such as unwanted $Z$ rotations due to imperfect control, which are not captured by stochastic error models. For the surface code, numerical studies have indicated threshold behavior even under coherent errors, but a rigorous proof of threshold existence is lacking. Here we prove that a positive threshold for coherent $Z$-rotation errors exists for quantum low-density parity-check codes with a bounded number of logical qubits, including the surface code and other topological codes. Specifically, we show that the maximum-likelihood Pauli recovery suppresses the entanglement infidelity exponentially in the code distance up to a prefactor linear in the number of physical qubits whenever the rotation angles lie below a constant that is independent of the code size. The proof combines Fourier analysis to retain the interference among the amplitudes of coherent errors with the cluster expansion of abstract polymer models. Our results expand the theoretical foundation of quantum error correction and offer a statistical-mechanical description of quantum error correction beyond stochastic errors.
发表机构
- The University of Tokyo（东京大学）
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