AI 中文总结
该研究针对带标记擦除的一维协变编码器，结合互补信道几何等方法，得出Charge-Haar码的擦除律与相变结论，证明局域保数砖电路的编码时间下界，确定扩散是对称约束量子编码的操作极限。
AI 中文摘要
量子纠错码的快速制备对可扩展量子存储器至关重要，但几何局域性与U(1)电荷守恒施加了不可避免的输运约束。我们结合精确的互补信道几何、电荷区Haar分析及门分辨的连通矩展开，研究带标记擦除的一维协变编码器。Charge-Haar码达到普适相邻电荷下界，误差为指数小量，给出精确的n^{-1/2}广延擦除律与尖锐的半擦除相变。对于局域保数砖电路，逻辑电荷的扩散强制了Ω(n²)的编码时间下界；我们还证明经典分量的O(n³)混合界，并将剩余全信道上界归约为源限制的低支撑算符扩散问题。这些结果确定了扩散是对称约束量子编码的操作极限，为其精确形成时间建立了研究途径。
英文摘要
Fast preparation of quantum error-correcting codes is essential for scalable quantum memories, but geometric locality and $U(1)$ charge conservation impose an unavoidable transport constraint. We combine exact complementary-channel geometry, charge-sector Haar analysis, and a gate-resolved connected-moment expansion to study one-dimensional covariant encoders under flagged erasure. Charge-Haar codes attain the universal adjacent-charge lower bound up to exponentially small corrections, yielding an exact $n^{-1/2}$ extensive-erasure law and a sharp half-erasure transition. For local number-conserving brickwork circuits, diffusion of the logical charge enforces an $Ω(n^2)$ encoding-time lower bound; we also prove an $O(n^3)$ mixing bound for the classical component and reduce the remaining full-channel upper bound to a source-restricted low-support operator-spreading problem. These results identify diffusion as an operational limit on symmetry-constrained quantum coding and establish a route to its exact formation time.
