发表机构
Technical University of Munich; BMFTR Research Hub 6G-Life, Germany; Munich Centre for Quantum Science and Technology (MCQST); Munich Quantum Valley (MQV); Cluster of Excellence CeTI, Technical University of Dresden; Centre for Quantum Technologies, National University of Singapore; Nanyang Quantum Hub, Nanyang Technological University（慕尼黑工业大学; 德国 BMFTR 研究枢纽 6G-Life; 慕尼黑量子科学与技术中心; 慕尼黑量子谷; 德累斯顿工业大学卓越集群 CeTI; 新加坡国立大学量子技术中心; 南洋理工大学南洋量子枢纽）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出算子代数框架，结合辛几何与张量网络，为量子电路容错性建立充要条件，并导出SDP测试、距离矩阵及LP上界，应用于Floquet码和flag提取电路，验证了动态编码与距离界限。
AI 中文摘要
我们发展了一个算子代数框架，用于分析量子电路中的容错性，并在任意给定噪声模型下建立了容错性的充分必要条件。对于指定的电路族和噪声模型，我们推导出一个基于半定规划（SDP）的测试，其不可行性证明不存在任何恢复映射能够恢复其预期操作。对于非自适应稳定子电路，利用辛几何分析其组件的稳定子对称性，我们获得了闭式检测矩阵和逻辑效应矩阵，这些矩阵以代数方式刻画了电路距离。应用于一个有限的Hastings--Haah蜂窝Floquet码电路，这些矩阵在包含测量误差的Pauli噪声下，证明了两个逻辑量子比特的动态编码和距离为四。利用这些矩阵，我们进一步推导了电路权重枚举器的MacWilliams恒等式，并获得了电路距离的线性规划（LP）上界。输出码约束给出了额外的Singleton类上界。在独立同分布（i.i.d.）去极化噪声下，权重枚举器决定了译码器失败概率，并给出了有限电路伪阈值的上界。最后，我们通过将$t$-flag准则作为附加线性约束纳入距离界限LP，为由其CNOT顺序指定的标志综合征提取电路族推导了LP距离界限。一个解析的CSS Hamming码基准测试产生了由Chao--Reichardt单flag构造达到的紧致距离三界限。
英文摘要
We develop an operator-algebraic framework for analyzing fault tolerance in quantum circuits and establish necessary and sufficient conditions for fault tolerance under any given noise model. For prescribed circuit families and noise models, we derive a semidefinite program (SDP) based test whose infeasibility certifies that there does not exist any recovery map restoring its intended operation. For non-adaptive stabilizer circuits, analyzing the stabilizer symmetries of its components with symplectic geometry, we obtain closed-form detection and logical-effect matrices that characterize circuit distance algebraically. Applied to a finite Hastings--Haah honeycomb Floquet-code circuit, these matrices certify the dynamical encoding of two logical qubits and distance four under Pauli noise including measurement errors. Using these matrices, we then derive MacWilliams identities for circuit weight enumerators and obtain linear-programming (LP) upper bounds on circuit distance. Output-code constraints give additional Singleton-like bounds. Under independent and identically distributed (i.i.d.) depolarizing noise, the weight enumerators determine decoder failure probabilities and yield upper bounds on finite-circuit pseudothresholds. Finally, we derive LP distance bounds for families of flag syndrome-extraction circuits specified by their CNOT orderings, by incorporating the $t$-flag criterion as additional linear constraints to the distance-bounding LP. An analytical CSS Hamming-code benchmark yields a tight distance-three bound attained by the Chao--Reichardt one-flag construction.
Comments46 pages, 5 figures
