用转移矩阵建模量子错误检测
Modeling Quantum Error Detection with Transition Matrices
- Yale Quantum Institute & Department of Computer Science, Yale University（耶鲁量子研究所与计算机系，耶鲁大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出用转移矩阵精确描述稳定子码在电路级噪声下的量子错误检测动力学，支持后选择与粗粒化，并证明降低检查频率可提升前导阶效率，在$[[4,2,2]]$和$[[5,1,3]]$码上验证了与蒙特卡洛的一致性。
AI中文摘要：
我们构造了一个显式的转移矩阵，该矩阵精确描述了在电路级随机泡利噪声下稳定子码的码适配基态布居动力学。该矩阵以码适配基表示，捕获逻辑基标签与综合征扇区之间的概率流。对于量子错误检测（QED），我们通过将拒绝的结果聚合到一个吸收态来纳入后选择，从而使得单个转移矩阵能够表示一个完整的时钟周期，包括门操作、综合征提取和后选择。此外，我们刻画了协议对称性何时允许精确粗粒化到更简单、更具可解释性的模型。转移矩阵框架使得经典随机矩阵技术能够直接应用于分析多周期QED协议。首先，我们识别了稳态逻辑-only接受映射描述的前导阶障碍，并量化了由QED检查不完美性引起的接受泄漏注入，该泄漏导致了这一障碍。其次，我们将前导阶QED效率表示为检查频率的函数，用物理上有意义的参数表达，并证明较不频繁的检查在前导阶上提高效率。我们在$[[4,2,2]]$和$[[5,1,3]]$码上，在去极化门噪声和读出错误下评估了该框架：转移矩阵预测与百万次射击的Stim蒙特卡洛在报告的置信区间尺度上一致，矩阵计算结果展示了首周期瞬态和区间依赖的效率。更广泛地，转移矩阵形式为QED分析提供了分析基础，并为理解码在现实噪声下的行为提供了可解释的工具。
英文摘要：
We construct an explicit transition matrix that exactly describes code-adapted basis-population dynamics of stabilizer codes under circuit-level stochastic Pauli noise. The matrix is expressed in a code-adapted basis that captures probability flow between logical-basis labels and syndrome sectors. For quantum error detection (QED), we incorporate post-selection by aggregating rejected outcomes into an absorbing state, so that a single transition matrix represents a full clock cycle of gates, syndrome extraction, and post-selection. Additionally, we characterize when protocol symmetries permit exact lumping to simpler, more interpretable models. The transition matrix framework enables direct application of classical stochastic-matrix techniques to analyze multi-cycle QED protocols. First, we identify a leading-order obstruction to a stationary logical-only accepted-map description, and quantify the accepted leakage injection from QED-check imperfections that causes it. Second, we express leading-order QED efficiency as a function of check frequency in terms of physically meaningful parameters, and prove that less frequent checks improve efficiency to leading order. We evaluate the framework on the $[[4,2,2]]$ and $[[5,1,3]]$ codes under depolarizing gate noise and readout errors: transition-matrix predictions agree with million-shot Stim Monte Carlo at the scale of the reported confidence intervals, and matrix-computed results illustrate the first-cycle transient and interval-dependent efficiency. More broadly, the transition-matrix formalism provides both an analytical foundation for QED analysis and an interpretable tool for understanding code behavior under realistic noise.
